Mr. Franken’s announcement on Thursday was a jarring end to an improbable political career in which the senator, a founding writer and performer on “Saturday Night Live,” narrowly won a seat in 2008 and offered Democrats a crucial vote needed to advance the Obama administration’s agenda, including the Affordable Care Act.

Mr. Franken cut a serious figure in the Senate, where he sought to stifle his sense of humor as he dived into meaty policy issues like electronic privacy and telecommunications mergers. Some viewed him as a potential contender for the presidency in 2020.

The accusations against him began last month when Leeann Tweeden, a radio news anchor in California, accused Mr. Franken of forcibly kissing and groping her on a U.S.O. tour in 2006. Several women also said Mr. Franken groped them as he posed with them for photographs, mostly before he became a senator.

Over the last three weeks, Mr. Franken has repeatedly apologized for his behavior, although he has also challenged some of the accusations of impropriety. Until Wednesday, he had said he would remain in his job, but his Democratic colleagues in the Senate made clear this week that his apologies and admissions were not sufficient.

Senator Tim Kaine, Democrat of Virginia, said that Mr. Franken’s resignation crossed “a cultural Rubicon” and set a new standard by which future senators would be judged.

“Now it’s clear that behavior of this kind before you take office is something that the body should take seriously — and it should be evenhanded whether you are a Democrat or a Republican,’’ Mr. Kaine said.

The Senate was somber as Mr. Franken delivered his speech on Thursday. His staff and family, including his wife, Franni Bryson, watched from the gallery above. About 20 Democrats and, apart from Senator Dan Sullivan, Republican of Alaska, who was presiding, just one Republican — Senator Jeff Flake of Arizona, an ardent critic of Mr. Trump — arrived in the chamber to hear his remarks.