A new CNN survey showed that about as many Democrats under 50 would be upset or dissatisfied with Mr. Biden as the nominee as they would be enthusiastic. And among those older than 65, views were even starker about Mr. Sanders: just 23 percent said they’d be enthusiastic about him while 33 percent said they’d be upset or dissatisfied.

Mr. Sanders has tried to bolster his standing with older voters, and lessen their ardor for Mr. Biden, by trumpeting his support for Social Security and highlighting the former vice president’s past willingness to consider cuts to the program — a contrast Sanders supporters believe is vital given Mr. Trump’s suggestion this week that he’d pursue entitlement trims.

Interviews with Sanders supporters at his events in New Hampshire and at the King Day gathering in South Carolina revealed a group of progressive activists who were as dedicated to him as they were in 2016 — and who were uneasy about his rivals, especially Mr. Biden. That was borne out in a new poll of New Hampshire primary voters this week from Suffolk University, which indicated that nearly a quarter of the Vermont senator’s supporters would not commit to backing the party’s nominee if it was not Mr. Sanders.

That number could drop by November if Mr. Sanders does not win the nomination: research shows that most of Mr. Sanders’s supporters eventually rallied to Mrs. Clinton against Mr. Trump. Yet it would not necessarily happen easily, especially if Mr. Sanders’s supporters believe he’s been treated unfairly by the party.

Many Sanders supporters who said they would grudgingly support one of his rivals against Mr. Trump quickly added that that’s all they’d do, ruling out doing the volunteer work that is the lifeblood of all campaigns.

“I just couldn’t morally,” Laura Satkowski said, explaining why she would not canvass or make phone calls on behalf of Mr. Biden. “I don’t like his policies.”

Some pro-Sanders households are mixed.

Michelle McKay and her partner, Bill Davis, came to the South Carolina State House from their home in Raleigh, N.C., she wearing a vest festooned with Sanders buttons, to show their support for their candidate.