Some of his advisers said in interviews that they were privately nervous that his recent gaffe spree would become cemented into the larger narrative of the presidential race. They also say that Mr. Biden faces an unfair double standard.

Yet there is a real political risk for Mr. Biden. Some party activists have already been worried that, at 76, he may be too old to go toe-to-toe with Mr. Trump, who is 73, and win. If the accumulation of verbal missteps continues, some Democrats say, it will eventually sow doubts about what many primary voters believe is Mr. Biden’s biggest strength: that he is best positioned to beat Mr. Trump.

“I have to say, he just doesn’t seem to have his heart in it or the energy for the slog this time,” said Chris Henning, the chairwoman of the Greene County Democratic Party in Iowa, who caucused for Mr. Biden when he ran for president in 2008 and is uncommitted this cycle.