“The first-blush analysis is inadequate,” Ms. Gillibrand said in an interview. “This is what makes me the best person to take on Trump — electability. Experience. Track record.”

“I’m the most elect ——” she stopped. “I have the type of experience they’re looking for.”

At this early stage of the Democratic presidential primary, much of the discussion among voters has focused on the singular desire of defeating Mr. Trump, and selecting a nominee who’s best suited to that task. But while that line of thinking has largely been associated with well-known veteran male politicians, particularly former Vice President Joseph R. Biden Jr., the women running in the historically diverse Democratic field, several of whom have a demonstrated track record of winning over Republican voters, have been telling anyone who will listen that they, too, are equipped to beat the president.

In addition to Ms. Gillibrand, Senator Amy Klobuchar of Minnesota has drawn on her electoral success in red counties to position herself as a bridge-builder in increasingly polarized times. And Senator Elizabeth Warren of Massachusetts — who soundly defeated a popular Republican incumbent in her first election — has focused recently on addressing concerns that she’s simply an “ideas candidate,” combining her rhetoric about economic inequality with a more explicit pitch on her ability to beat Mr. Trump. (A fourth leading female candidate, Senator Kamala Harris of California, has enjoyed most of her success thus far in Democratic strongholds.)

As they now campaign for president, they are encountering some of the same misogyny that Mrs. Clinton faced when she ran in 2016. They are running up against assumptions voters and pundits have about what presidential leadership looks like, battling a presidential archetype where men are the only touchstones.