Representative Marc Molinaro has spent his decades of public service building a reputation as a particular brand of New York Republican: a measured and courteous pragmatist more interested in responsible governing than in ideological battles.

But that reputation is being tested as Mr. Molinaro, a first-term congressman, seeks re-election against Josh Riley, a Democrat, in a rematch from 2022 that has turned into one of the most hostile, and consequential, House races in the country this year.

In a blistering debate last week during which the candidates traded charges of lying and corruption, Mr. Molinaro sought to tie Mr. Riley to Democrats’ border policies, blaming them for violent crimes including a rape in Albany and the murder of a family in Rochester.

“Why?” Mr. Molinaro said, his face flushing angrily. “Because they use the legal argument that Josh Riley made to surrender the border.”

Mr. Riley, a lawyer and policy analyst, responded quickly.

“Every one of those incidents that he’s talking about and putting on TV — that happened on his watch. He’s in Congress,” he said, noting that Mr. Molinaro had joined other House Republicans, at former President Donald J. Trump’s urging, in rejecting bipartisan immigration legislation supported by the Border Patrol.