So with Mr. Biden leading the way, Democrats have spared no effort in the race. Money has poured in to support Ms. Hansen, including some from states like New York and California: She reported raising more than $300,000 over the last three weeks, several times the normal budget of a Delaware legislative race. Ms. Hansen’s opponent, John Marino, a real estate broker and a retired New York City police officer, raised less than one-sixth of that.

Democratic voters outnumber Republicans in the district, but turnout is unpredictable in special elections, and Mr. Marino nearly pulled off an upset win in 2014, the last time the seat was on the ballot. He lost by two percentage points to Bethany Hall-Long, a Democrat who gave up the seat after winning a race for lieutenant governor in November.

In addition to Mr. Biden, Ms. Hansen has been joined on the trail by Delaware’s Democratic governor, John Carney; its two Democratic senators, Thomas R. Carper and Chris Coons; and Martin O’Malley, a former Maryland governor and presidential candidate. Labor unions and an outside group, First State Strong, which has not disclosed its donors, have also spent hundreds of thousands of dollars supporting Ms. Hansen.

Mr. O’Malley called the investment in Delaware a sign that Democrats were “waking up to the reality that we’ve got to rebuild our party” at the state level. He said they had grown too complacent about state-level elections during Mr. Obama’s administration and suffered staggering losses as a result.

Republicans now control 33 governorships and have full control of the government — holding the governor’s offices and the legislatures — in 25 states.

“Sometimes, as Democrats, we kid ourselves into thinking the only office that matters is the presidency and the only legislature that matters is the United States Senate,” Mr. O’Malley said. “But you can’t be a national party if that’s the warped perspective you maintain.”