President Biden on Sunday made his first visit to the border since taking office, arriving at a city swamped by migrants amid a historic surge in illegal immigration and anger from both parties about how he is handling it.

In a brief visit to El Paso’s busiest crossing and a migrant services support center, Mr. Biden acceded to demands by Republicans that he make the trip he had not taken for two years.

But as he arrived in El Paso, he found himself under siege from all sides.

Democrats and human rights activists condemned his new enforcement plan as a “humanitarian disgrace.” Republicans blasted his delay in coming to a border they say is “wide open” to illegal immigration. And Mexican officials — who are preparing to welcome him to a summit of North American leaders on Monday — warned that his proposals would cross a “red line” for them.

The number of migrants apprehended trying to illegally cross the 2,000-mile border with Mexico has hit record highs. In the 12 months leading up to last October, the Border Patrol encountered 1.7 million migrants trying to cross illegally, the highest number since 1960. Officials said overall crossings had dipped some during the holiday season in December, but they said they expected the numbers to rise again quickly in the coming months.

The surge in migrants has reached deep into the United States. Cities as far away as New York and Washington are struggling to provide services to the growing numbers who are arriving, some at the behest of Republican governors in Florida and Texas, who have bused or flown the migrants out of their states.

In El Paso, a record-breaking swell of migration from across Central and South America has made the city one of the most vivid symbols of the decades-long breakdown in America’s immigration system. Desperate people, often with small children, spend cold nights on park benches, with no legal status and nowhere to go after making the brutally dangerous trek north in the hopes of finding refuge.

The question of what to do with them — Accept them? Detain them? Send them home? — has become one of the most polarized political debates in the United States. And Mr. Biden has not found a solution as the situation in El Paso and communities along the border has worsened during his presidency.

On Sunday, Mr. Biden met with Border Patrol officers, members of Congress and local officials at the Bridge of the Americas Port of Entry, El Paso’s busiest crossing, which is set to receive $600 million from the president’s infrastructure law. He also made an unannounced stop along the 18-foot border wall that separates El Paso from Juárez, Mexico, talking to Border Patrol agents as he strolled along a dirt road on the American side.

Asked later what he learned, Mr. Biden said: “They need a lot of resources. We’re going to get it for them,” apparently referring to the Border Patrol agents he just met with.

Mr. Biden also toured the El Paso County Migrant Services Center and talked to local business owners.

Mr. Biden did not visit areas where migrants have been sleeping outdoors in El Paso. Hours before he landed, streets were relatively quiet. But the mood was tense mere steps from the Bridge of the Americas in El Segundo Barrio, which has become the epicenter of the migrant crisis in the city.

“I feel like she protects me, so I sleep right here next to her,” Mr. Cardenas said, his eyes scanning the image next to him. But on Sunday he was praying for a different kind of protection. “I hope Biden can offer us some help, too. We need it desperately.”

Jesús Ramirez, 19, another Venezuelan who traveled with Mr. Cardenas from Panama, also arrived on Dec. 31. He said he does not regret the risks he took during a grueling three-month trek to the U.S. border. He saw people die along the way, even children, he said, closing his eyes. He still has nightmares about the dead bodies he saw in the Darién Gap, a dangerous and roadless land bridge that connects Colombia and Panama.

On the Mexican side on Sunday afternoon, fewer than a dozen migrants, the majority from Venezuela, appeared defeated, staring across the Rio Grande, where they were welcomed by a newly installed fence, barbed wire and a row of heavily armed Texas National Guard members.

By Sunday, a group of men had turned the small gap behind a fence of Sacred Heart Church into rudimentary accommodations. They turned a pile of sheets, some of them donated by the Red Cross, into beds, next to worn-out Bibles they carried from home. John Cardenas, a native of Venezuela who arrived in El Paso on Dec. 31, prefers to sleep next to a large statue of the Virgin of Guadalupe.

Denyerlin Chirino, 24, stood next to her 3-year-old, Chantal, and a group of fellow travelers. Ms. Chirino surrendered at a port of entry on Dec. 23, she said, but was quickly expelled. She had been trying to reach family in Orlando.

“I came here today because I heard the U.S. president is visiting the border,” she said, adding, “I’m hoping against hope that he does something for us.”

Moments after Ms. Chirino had spoken with The New York Times, a caravan of Mexican state police vehicles raced across the river and stepped on their brakes when they spotted the group of migrants. Ms. Chirino grabbed her daughter and took off running.

Responding to mounting criticism of his handling of the border, Mr. Biden on Thursday announced a crackdown on illegal border crossings paired with some new avenues for migrants to immigrate legally. Mr. Biden has leaned on several Trump-era immigration policies, including for a time one that required migrants to wait in squalid camps in Mexico while their asylum applications were processed.

As a candidate, Mr. Biden assailed President Donald J. Trump’s handling of the border, calling it inhumane and ineffective. But as Mr. Biden heads to Mexico City for two days of meetings that will be dominated by the issue of immigration, his critics have only ramped up their assault on his policies.

“They flout refugee law & will unleash more suffering at the border, w/ disparate harm to Black, Brown & Indigenous asylum seekers,” Heidi Altman, the policy director at the National Immigrant Justice Center, a liberal human rights group, said on Twitter.

The criticism from the same groups that fiercely opposed Mr. Trump’s policies has infuriated Mr. Biden and his aides, who say the comparison is unfair and wrong.

“Well, obviously, we take a different view,” said John F. Kirby, a top spokesman for the National Security Council, noting the demand to balance the need for “legal pathways to entry” and ensuring that “illegal migration is curbed.”

The president’s aides also braced for renewed attacks from Republicans who have spent months accusing the administration of being too weak on the border. In an opinion article published before the president’s trip, Senator Shelley Moore Capito, Republican of West Virginia, chided Mr. Biden for waiting almost two years to visit.

“The president neglecting to visit the southern border — during a time when we are facing record illegal crossings and there is a clear crisis — would be the equivalent of our commander in chief not visiting the Pentagon during a military operation,” she wrote.

Republicans have argued that Mr. Biden’s desire to be more “humane” along the border was an open invitation to increased migration, and they have assailed what they said was the administration’s continued release of many migrants into the United States while their immigration cases are litigated.

Upon arriving in El Paso on Sunday, Mr. Biden was greeted by Gov. Greg Abbott of Texas, who handed him a letter. A news release from the governor’s office said that the letter said, “This chaos is the direct result of your failure to enforce the immigration laws that Congress enacted.”

The question of what to do along the southern border in places like El Paso will be at the center of the discussions when Mr. Biden arrives for meetings with President Andrés Manuel López Obrador of Mexico and Prime Minister Justin Trudeau of Canada.

In an interview before Mr. Biden’s departure on Sunday, Mexico’s top official for North American relations bluntly rejected a proposal by the Biden administration that would automatically deny asylum to migrants who have traveled through Mexico without seeking refuge in that country first. A version of that idea, known as a “safe third country” policy, was first proposed by Mr. Trump in 2019.

“The safe-third-country idea is a red line for us,” said Roberto Velasco, the Mexican Foreign Ministry’s chief officer for North America. “It would overwhelm the system.”

Finding ways to cooperate with Mexico on migration is a key goal for Mr. Biden on his visit to the country. In his remarks on Thursday, the president announced that Mexico had agreed to accept up to 30,000 migrants each month from Haiti, Venezuela, Cuba and Nicaragua if they tried to cross into the United States illegally and their asylum claims were rejected.

But longtime observers of the relationship between the two countries said there were limits to what Mexico would accept.

“Mexico cannot guarantee the security of those fleeing violence,” said Ana Lorena Delgadillo, the director of the Foundation for Justice, an organization that supports migrants in Mexico. “Mexicans are fleeing violence in their own communities. How are we going to protect others if we cannot protect our own?”

While the current Mexican administration has pushed hard to stem the stream of migrants traveling across Mexico and into the United States, security experts said the government had done less to stop drugs from flowing north — another planned topic of discussion at the summit this week.

David Shirk, the director of the Justice in Mexico Program at the University of San Diego, said that “what we have seen in the last three years is essentially the Mexican government shrugging its shoulders.”

But on Thursday, the Mexican military detained Ovidio Guzmán López — the son of the infamous drug lord Joaquín Guzmán Loera, known as “El Chapo” — who is believed to be a leader of the Sinaloa drug cartel that his father once headed.

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The response from the cartel was swift, and violent, with vehicles lit on fire across the capital city of Culiacán, according to law enforcement officials and videos posted on social media. Videos showed armed groups controlling checkpoints near the city, carrying high-powered machine guns capable of taking down helicopters.

At the summit on Monday and Tuesday, the three leaders are also expected to clash over a sharp rise in the number of formal trade disputes since the United States-Mexico-Canada Agreement went into effect in 2020. Several conflicts loom large. The United States and Canada have protested Mexican energy policies that benefit Mexican firms over their companies. Mexico also plans to ban genetically modified corn, to the frustration of the United States. And the other countries have complained about how the United States is interpreting the trade agreement’s rules for making cars, arguing that Washington is favoring American products over their own.

American officials have tried to emphasize the benefits that its neighbors can expect to see from the investments, given that the countries’ economies are so intertwined. The United States, Canada and Mexico trade $3 million of goods every minute, and they share heavily integrated supply chains, in which parts and raw materials made in one country are often used in the factories of the others.

The trade issue is particularly important for Mr. Trudeau, who has served as Canada’s leader since 2015 but has recently become a more polarizing figure in his country.

“There’s been some discussion, because he’s been in power for quite a while, about who the potential successor might be,” said Andrew McDougall, a political scientist at the University of Toronto. “But there isn’t any obvious candidate right now and, in the medium term, it doesn’t seem like he’s going anywhere.”