Republicans also hope that the more that Democrats talk about impeachment, the more opening they will have to cast it as illegitimate.

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“It’s not lost on people that many of these candidates were actually on the record supporting impeachment before anyone was even talking about Ukraine,” said Andy Surabian, a Republican strategist and former Trump White House official.

The debate, held outside Columbus, Ohio, allowed Republicans to try to make their case in a state that Mr. Trump won by eight percentage points in 2016 and has many of the middle-class, blue-collar voters that he hopes to turn out again in next year’s election. To underscore the point, the Trump campaign took out a full-page ad in the Columbus Dispatch on Tuesday that blared, “Democrats Will Kill Ohio Jobs … President Trump’s Fighting for the Economy.”

The president remains historically unpopular, however. The Gallup presidential approval poll has measured public support for his job performance at below 50 percent for his entire presidency. To overcome this skepticism, Republicans have attempted to define the Democrats as out of touch and too far to the left of where most Americans’ views are. Some of the Democratic candidates, like former Vice President Joseph R. Biden Jr. and Mayor Pete Buttigieg of South Bend, Ind., hold moderate and liberal views, while Senators Elizabeth Warren of Massachusetts and Bernie Sanders of Vermont are generally more progressive.

Senator Marco Rubio of Florida on Wednesday expressed the kind of binary argument that Republicans hope prevails on voters. “Last night’s debate was a reminder that while you may not always agree with everything Trump says or does, the leading alternatives to Trump mock people with traditional beliefs & support socialism, gun confiscation & free government funded health care for unlawful immigrants,” he wrote on Twitter.

With Ms. Warren the target of sustained attacks from the other Democrats onstage, Mr. Trump and his Republican allies moved to frame the race around the question of whether voters want a strongly liberal president and government in 2021.

On Tuesday night, after Mr. Sanders picked up the support of two high-profile liberals in the House Democratic caucus — Representatives Alexandria Ocasio-Cortez and Ilhan Omar — Republicans jumped at the chance to make them a focal point in the morning-after debate analysis. The two freshmen are a favorite target of conservatives, who have caricatured them as the true leaders of their party.