Hundreds of miles from New York City, Republicans desperate to protect a House seat in Central New York aired television ads vilifying Mayor Eric Adams, splicing in images of migrants sleeping on city sidewalks.

In the Hudson Valley, a Republican congressman homed in on New York City during a debate this month, highlighting crimes committed by migrants in Manhattan. And on Long Island, a Democratic hopeful struck a similar note, vowing to secure the southern border because the spillover from New York City had the suburbs “feeling the migrant crisis almost every day.”

Few cities have borne the burden of the migrant crisis like New York, which has received more than 210,000 asylum seekers over the past two years. But the political fallout of the crisis has spilled into competitive House races across the state. Republicans have used New York City to illustrate the disorder that they say has been wrought by Democratic immigration policies, with Democrats responding by adopting a tougher approach to border security.

The city’s migrant crisis has also come up in the presidential race. Former President Donald J. Trump, the Republican nominee, claimed at a rally this month that migrant criminals had “taken over Times Square,” exaggerating recent reporting by The New York Post about migrant gang activity in Manhattan.

In doing so, Mr. Trump typified his party’s strategy of intertwining the migrant influx with two other pressing issues for voters: crime and high living costs.

That tactic has been frequently deployed by Republicans in contested House races in New York that could help decide control of Congress, with New York City a frequent target in debates, stump speeches and attack ads.