“At one point, he’s talking about people in communities like mine listening to record players — I don’t remember the last time I saw a record player,” Mr. Booker said, adding, “There are definitely moments when you’re listening to Joe Biden and you just wonder.”

Mr. Biden and his campaign surrogates have taken umbrage at suggestions that he has slipped in his political acuity, and on Friday they pushed back particularly hard on Julián Castro, the former federal housing secretary, for having bitingly questioned Mr. Biden’s recall of his own policies in the debate. Mr. Castro, who is polling near the bottom of the field, was the only candidate to attack Mr. Biden in such strong terms, drawing backlash that illustrated why other candidates had shunned that approach.

Asked by a reporter Friday whether he would release his medical records to address “concerns,” Mr. Biden, 76, said he would do so before votes are cast.

“What the hell concerns, man, you want to wrestle?” he said lightheartedly. He continued, “When I get the next physical. Look, I’ll release my — before there’s a first vote, that’s — I’ll release my medical records. There’s no, I mean there’s no reason for me not to release my medical records.’’

In Mr. Biden’s camp, there is a persistent sense that his rivals and the news media are underestimating him and giving him too little credit for the blocs of support he has already claimed. Whatever the flaws in Mr. Biden’s performance in Houston, there was no exchange in which an opponent obviously routed him, as Senator Kamala Harris of California did in the first debate and Mr. Booker did in the second.

He has faced many controversies throughout the campaign, his allies note, and he is still in the lead — a reflection, they argue, of the good will he enjoys from rank-and-file Democrats who feel that they already know him and who see him as best positioned to defeat Mr. Trump.