Many moderate Democrats see the field’s current front-runner, Joseph R. Biden Jr., the 76-year-old former vice president, as a safer option than Ms. Warren and other candidates. But Mr. Biden’s lead in the polls is partly based on strong name recognition, and his recent gaffes and middling debate performances have raised questions about whether he has the agility to defeat Mr. Trump.

Many voters interviewed are now wrestling with whether to elevate a candidate who captures their imaginations, and progressive ambitions, or to rally more cautiously behind a Democrat who they perceive as having a better chance of building a broad coalition of Democrats, independents and disaffected Republicans to fulfill their most urgent goal: ejecting Mr. Trump from the White House.

The Massachusetts senator’s top campaign aides are acutely aware of their challenge on questions about Ms. Warren’s viability. They are taking a series of steps to allay the concerns, perhaps most notably arming her in the last debate with the talking point that conventional wisdom also suggested that both Mr. Trump and former President Barack Obama were risky nominees because they broke from the traditional commander-in-chief mold. After the debate, Warren aides blasted clips of that remark from her social media accounts.

But even after Ms. Warren turned in two well-received debate performances, a Quinnipiac survey showed she had not made gains on the question of who has the best chance to beat Mr. Trump: Just nine percent said she did, while 49 percent pointed to Mr. Biden.

In an interview before a town hall meeting in western Iowa last week, Ms. Warren, acknowledging the questions about her candidacy, said there was only one overarching way to quiet the skeptics.

“Nothing will overcome people’s worries more than success,” she said.

But Ms. Warren also demonstrated that she was still uncertain about how to address Mr. Trump’s taunts about the Native American heritage she once claimed. Her attempt to prove that ancestry with a DNA test last year drew fierce criticism from the right and left as well as some Native American groups; she stood by the DNA test for months, then apologized for it and the claims.