Privately, some Democratic officials in Minnesota said they felt uneasy not calling for his resignation, worried that it would expose them to accusations of hypocrisy. This week, some national Democrats have publicly questioned the response at the time to Bill Clinton’s sexual relationship with an intern while he was president, with Senator Kirsten Gillibrand of New York now saying that Mr. Clinton should have resigned.

But they also tried to draw a distinction between Mr. Franken’s behavior — which they saw as isolated — and the pervasive, longstanding patterns of sexual harassment that other elected officials were accused of.

One of Mr. Franken’s close allies, Senator Amy Klobuchar, also a Minnesota Democrat, said that she “strongly” condemned his behavior and that the Senate Ethics Committee should investigate it, but she has not called for him to step down. The editorial board of The Star Tribune in Minneapolis, which only days ago called for the resignations of the state legislators accused of sexual harassment, including unwanted advances and unsolicited text messages with sexual content, wrote: “Franken, so far, is accused of one incident that occurred before he took office. That does not excuse or mitigate the gravity of his conduct, which was despicable, but the story was still developing as this editorial was being written.”

On Friday, several women who had worked on Mr. Franken’s staff released a statement in his defense. “Many of us spent years working for Senator Franken in Minnesota and Washington,” the statement said. “In our time working for the senator, he treated us with the utmost respect. He valued our work and our opinions and was a champion for women both in the legislation he supported and in promoting women to leadership roles in our offices.”

A native of Minnesota, Mr. Franken ran for the Senate in 2008 after a career as an entertainer, liberal radio host and author of books with titles like “Lies and the Lying Liars Who Tell Them: A Fair and Balanced Look at the Right.” He is eligible to run for a third term in 2020.

Minnesota is led by one of the most liberal governors in the country, Mark Dayton, who has repeatedly clashed with the Republican-controlled State Legislature. Hillary Clinton won the state by a razor-thin margin in 2016.