The seriousness of the campaign’s response seemed to elevate a nonsensical proposal. “That is a danger,” said Dan Pfeiffer, a former senior adviser to Mr. Obama. “You have to take the threat of Trump becoming president seriously, but you shouldn’t treat him as a serious person.”

At the same time, Mrs. Clinton’s attempts at poking Mr. Trump have felt tame compared with what he has unleashed. The day after he accused her husband of rape on Fox News this month, she told CNN, “I have concluded he is not qualified to be president.”

Her aides were exuberant that Mrs. Clinton, a cautious candidate who for weeks had demurred when asked if Mr. Trump was up to the job, had finally publicly declared him unfit. But to many Clinton allies, it felt like a tepid tactic from yesteryear.

“Sometimes, you get the feeling that they’re in a professional boxing match and he’s in a street fight, and they’re coming in with their gloves on,” said the Rev. Al Sharpton, expressing dismay over the Clinton operation’s apparent lack of appetite for combat.

“This is a street fight with a guy with a razor and a broken Coca-Cola bottle,” he added, “and you’ve got to fight him like that.”