Democrats backing Hillary Clinton, nervously eyeing Senator Bernie Sanders’s growing strength in the early nominating states, are turning to a new strategy to raise doubts about his candidacy, highlighting his socialist beliefs to warn that he would be an electoral disaster who would frighten swing voters and send Democrats in tight congressional and governor’s races to defeat.

It is a scenario many Democrats long dismissed as even remotely plausible: the 74-year-old Mr. Sanders, a registered independent who self-identifies as a democratic socialist, as their nominee. But the possibility of his defeating Mrs. Clinton in Iowa and New Hampshire next month has prompted some of her prominent supporters to discuss how they could attack Mr. Sanders if his candidacy began to look less like a threat and more like a runaway train: calling him unelectable and warning Republicans would have a field day if he were the Democratic nominee.

“Here in the heartland, we like our politicians in the mainstream, and he is not — he’s a socialist,” said Gov. Jay Nixon of Missouri, who is term-limited and working to elect a Democratic successor. “He’s entitled to his positions, and it’s a big-tent party, but as far as having him at the top of the ticket, it would be a meltdown all the way down the ballot.”

And after months of ignoring Republican cheerleading for Mr. Sanders, Mrs. Clinton’s campaign has started aggressively highlighting how much the opposition is openly providing him aid and comfort — mostly recently in a new ad by Karl Rove’s group American Crossroads that echoes Mr. Sanders’s attacks on Mrs. Clinton’s ties to Wall Street.