For years, President Biden has had a ready retort for the naysayers who have questioned his facility and fitness to run for president again at age 81 and to serve until he is 86. “Watch me,” he has said.

But in the days since tens of millions of Americans watched him fumble Thursday’s debate in real time, Mr. Biden has essentially adopted a new line: Trust me.

“Folks,” he said at a New York fund-raiser the next night, “I would not be running again if I did not believe with all of my heart and soul that I can do this job.”

It’s a cliché of political strategy that smart campaigns meet voters where they are. That typically means fashioning a strategy that taps into the public’s pre-existing feelings, rather than seeking to change how the electorate perceives matters.

Yet the trouble for the president is that even on the eve of his faltering debate, a New York Times/Siena College poll showed that 69 percent of voters — and 55 percent of Biden voters — saw Mr. Biden as too old to be an effective president. It is not a new concern: Nearly two years ago, a strong majority of Democratic voters said they wanted a new standard-bearer.

Now those persistent concerns from everyday Americans are being echoed publicly by many in the Democratic Party’s pundit class and privately by lawmakers, donors and strategists. They are worried about losing a 2024 campaign against former President Donald J. Trump, whom many view as an existential threat to the nation.