Republicans successfully made crime the defining issue of the 2022 midterm elections in New York, fanning fears about public safety to rout suburban Democrats and help secure the party its House majority.

Barely a year later, as another critical election season begins to take shape, they appear to be aggressively testing a similar strategy, hoping that the state’s growing migrant crisis will prove as potent a political force in 2024.

The rapid arrival in New York of more than 100,000 asylum seekers is already wreaking havoc on government budgets, testing the city’s safety net and turning Democratic allies against one another. Now, otherwise vulnerable Republicans in a half dozen closely watched districts have begun grabbing onto all of it as a lifeline to portray Democrats as out of touch and unable to govern.

“This is a crisis of their own making,” said Representative Mike Lawler, a Republican fighting to hold a suburban district Mr. Biden won by 10 points.

“It’s very similar to cashless bail,” Mr. Lawler said. “When you create a sanctuary city policy that invites migrants to come regardless of their status, you are going to get a lot of people coming, and now they can’t handle the influx.”

Hearing the same echoes, Democrats are determined not to be caught flat-footed as they were a year ago. From the suburbs of Long Island to here in the Hudson Valley, their candidates are spending late summer openly clashing not just with Republicans who say they are to blame, but also with their own party leaders, including President Biden.

In one of the most closely watched contests, Representative Pat Ryan, the lone frontline Democrat to survive the Republican suburban demolition last year, has teamed up with two Republicans to demand that Mr. Biden declare a state of emergency, and broke with his party to support a bill to discourage schools from sheltering migrants.

“The No. 1 thing I learned as an Army officer: When in charge, take charge,” Mr. Ryan said in an interview. “We are in a crisis, the president is in charge, and he and his team need to take charge.”

He is far from alone. Josh Riley, a Democrat who is trying to flip a neighboring district, called the president’s aloofness on the issue “offensive.”

Mondaire Jones, a former Democratic congressman mounting a comeback attempt further down the Hudson, warned of “consequences at the polls” if his party does not step up.

And his primary opponent, Liz Whitmer Gereghty, said Democrats across New York should be responding in lock step. “It kind of feels like we’re not,” she said.

Both parties caution that the reality on the ground, where 2,900 migrants arrived just last week, is shifting too quickly for them to know exactly where the battle lines will be by next fall, when voters will also be weighing abortion rights and the criminal trials of former President Donald J. Trump, currently the leading Republican candidate.

Republicans have been using fears about immigrants pouring across the border for years with only mixed success. And unlike a year ago, Democrats are trying to go on offense, accusing Republicans like Mr. Lawler of engaging in demagogy and reminding voters that his party helped stall a major immigration overhaul in Washington that they say might have prevented the latest influx.

“Everybody understands this is a potential liability,” said Tim Persico, a Democratic consultant who oversaw the party’s House campaign operation last cycle. “I know there’s been a lot of finger pointing and kerfuffles, but there’s also pretty good evidence the mayor and the governor are trying to figure out how to solve this.”

Still, there is little doubt that New York, a city known as a bastion for immigrants, is in the midst of a challenge to its political system with few modern parallels. Privately, Democratic pollsters and strategists are beginning to use focus groups and polls to test possible defenses on an issue they view as a tinderbox capable of igniting new political fires, fast.

New York is housing roughly 59,000 asylum seekers a night because of a unique right-to-shelter mandate that dates back decades and is preparing to enroll some 19,000 migrant children in public schools this fall. An archipelago of temporary shelters has cropped up in hotels, parks and on public land, prompting increasingly raucous protests.

And Mayor Eric Adams has repeatedly warned of budget cuts as the cost of caring for the newcomers spikes into the billions of dollars — taxpayer money that Republicans are quick to point out could otherwise be used to help New Yorkers.

As the numbers keep climbing, Democratic leaders have been forced to choose from unpalatable policy responses.

Mr. Adams, for instance, has repeatedly demanded that Gov. Kathy Hochul force reluctant counties outside the city to help shelter migrants. But doing so would prompt fierce backlash in many of the communities Democrats need in order to win the House, and the governor, who was already blamed for Democrats’ 2022 losses, has refused.

On the other hand, any attempt by the city or state to drastically curtail the services it offers migrants would meet blowback from the left.

The governor and mayor — along with congressional Democrats as ideologically diverse as Representative Alexandria Ocasio-Cortez and Mr. Ryan — are united in demanding more help from Mr. Biden. But push too hard and they risk bloodying their party’s standard-bearer heading into an election year.

The White House did announce on Wednesday that it would dedicate personnel to help New York process work papers for asylum seekers and request additional federal funds from Congress to help the state. But Mr. Biden, who has to make his own national political calculations around immigration, appears to have little interest in taking a more visible role.

Voters are watching. A recent poll conducted by Siena College found that 82 percent of registered voters view the influx as a “serious” problem, and a majority said that the state had “already done enough” for the asylum seekers and should focus on slowing their arrivals. The same poll showed nearly every major Democrat, including Mr. Biden, underwater among suburban voters.

In many ways, those poor ratings have freed Democrats facing competitive races to distance themselves from their party in ways that telegraph to voters their understanding of the problem while differentiating themselves from Republicans’ more hard-line views on immigrants.

It is a tricky balancing act. At the same time Mr. Ryan is locking arms with Republicans to pressure his own party, he is also trying to shift responsibility onto Republicans and defend himself against their attacks for making the county he once led a “sanctuary” for undocumented immigrants.

“Where you really get yourself in trouble as an elected official is when you don’t listen,” Mr. Ryan said, adding: “For political purposes, the MAGA Republicans want divisions and chaos. They are not actually working to resolve problems.”

The task may be easier for challengers who are taking on Republican incumbents whom they can blame for failing to enact the kind of changes to the immigration system that could curb illegal border crossings, speed up the asylum system and eventually relieve pressure on New York.

“In my district, the one person sitting at the table to fix this problem is Anthony D’Esposito, and he is doing nothing,” said Laura Gillen, a Democrat seeking a rematch against Mr. D’Esposito, who represents the South Shore of Long Island. (He and other New York Republicans helped pass an aggressive but partisan border security bill in May.)

But Ms. Gillen, who wants to represent a district Mr. Biden won by 14 points, said the president deserved blame, too. She called a letter last week from his homeland security secretary critiquing New York’s handling of the migrants as “irresponsible.”

Mr. Riley is taking a similar “all our politicians are failing us” approach, knocking both Mr. Biden and Representative Marc Molinaro, his Republican opponent.

“Look, this is a federal problem and it requires a federal response, and I think President Biden needs to get his act together and help solve it,” he said.

It is too soon to know whether the approach is working. In Mr. Ryan’s district, the views of voters interviewed near a hotel housing migrants appeared to break down on familiar lines. Dozens of voters, when asked by a reporter, voiced dissatisfaction with how migrants had been bused up from New York City, but they disagreed on who was to blame.

“Not just the county but the country can handle this,” said Faith Frishberg, a Democrat, outside a waterfront restaurant in Newburgh. “Most of this failure is a failure to not address the immigration policy.”

But there may also be a distinct drawback over time.

Blaming Democratic leaders like Mr. Adams or Mr. Biden may be expedient short-term politics. But it risks reinforcing the notion that Democrats cannot govern — a potentially powerful boomerang effect in a state that has registered some signs of weariness of one-party rule in recent years.

Republicans already appear eager to reinforce it.

“I have not seen a less coordinated, less competent way of dealing with human lives,” Mr. Molinaro said. “I know the reporting today has become a little bit about how the president is pointing at the governor, the governor at the mayor. The story line is Democrat leaders are pointing at each other.”

Timmy Facciola contributed reporting from Newburgh, N.Y., and Luis Ferré-Sadurní from New York.