Some Democrats here remain conflicted about Mr. Franken’s resignation, announced after some of his colleagues in Washington called on him to leave over the accusations. And Republicans, who control the Legislature but have struggled in statewide races, sense an opportunity to pick off the seat, which they had expected was out of reach until at least 2020.

Next year’s ballot will also include an open race for governor — Mr. Dayton is not seeking another term — and an election for Senator Amy Klobuchar’s position. Ms. Klobuchar, a well-known Democrat, is expected to run for another six-year term.

“The opportunity for Minnesota Republicans is actually quite bright,” said Jennifer Carnahan, the chairwoman of Minnesota’s Republican Party. She added that President Trump’s narrow defeat here in 2016, four years after Mitt Romney lost the state by more than seven percentage points, indicated the most favorable political landscape for her side in more than a decade. “I think we’re definitely a key player when you look at the national map in a way that maybe we haven’t been in the past.”

Several prominent Republicans had already lined up for the governor’s race, and some might now consider challenging Ms. Smith for the Senate. Ms. Carnahan criticized Ms. Smith as a member of the “very far left” whose appointment “was calculated to appease the Democrats out in Washington.”

Ms. Smith, a longtime figure in Democratic politics and a former executive with the regional Planned Parenthood organization, spent most of her career behind the scenes before becoming lieutenant governor in 2015. Those who have worked with Ms. Smith described her as smart and able to work across party lines, but also politically savvy and committed to Democratic issues.