WASHINGTON — Republican leaders gave up hope on Thursday of reopening the government before the new year, leaving the border wall impasse to House Democrats as they assume the majority next week — and presenting Representative Nancy Pelosi with her first major challenge as speaker.

House Democrats, who take control on Wednesday, are weighing three approaches to getting funds flowing, none of which would include additional money for President Trump’s proposed wall along the southwestern border. Whichever path they choose, party leaders said they would vote promptly on Jan. 3, hoping to project the image of Democrats as a steadying hand in Washington even as Republicans try to blame Ms. Pelosi and her party for the shutdown and lax border control.

“We will vote swiftly to reopen government and show that Democrats will govern responsibly in stark contrast to this chaotic White House,” Ms. Pelosi said in a statement.

Ms. Pelosi is determined to prevent the shutdown brinkmanship from interfering with the Democrats’ assumption of power and her ceremony-soaked return to the speakership. But it appeared almost certain that the careful rollout of Democrats’ legislative agenda — including a sweeping anticorruption and voting rights bill — would be at least partly eclipsed by the funding crisis.