Sticking by Mr. Trump after the surfacing of a 2005 recording in which he could be heard speaking about women in vulgar terms and bragging about being able to get away with sexual assault would subject Republican candidates to devastating and, quite possibly, career-ending attack.

But disavowing Mr. Trump, whose supporters make up the largest share of the Republican base, risks alienating those voters, potentially a no less lethal choice for Senate candidates in key races. And if Mr. Trump loses to Hillary Clinton, as polls now indicate is likely, the loss of those Senate contests could be crushing for a party that was already teetering and counting on a chance to rebuild after the election. An NBC News/Wall Street Journal poll conducted just before the debate showed Mr. Trump’s support cratering, with Mrs. Clinton assuming an 11-point lead nationally.

Still, Mr. Trump has die-hard supporters who have shown they will stay with him through every controversy he has sparked or endured. His populist, outsider message may be at odds with his background and even with some of his policy proposals, but it has taken hold with many voters, particularly working-class Republicans who are disenchanted with the party’s elite and deeply unhappy with President Obama’s stands on health care and immigration.

“The establishment is trying to hold on to their power, and McCain is one of them,” Ms. Rupnik said.

That was a common theme in interviews with Trump supporters on Sunday in Arizona and New Hampshire, two states with close Senate contests. Many spoke witheringly of incumbent Republican lawmakers who have renounced their support for the party’s presidential nominee.