“I guess they used that argument to its end, and so they have to find someone new,” Ms. Ocasio-Cortez said in a brief interview. “And who else to use except folks like myself and other freshmen congresswomen? We’re least like them in every way possible, so I think it’s a potent symbol.”

But Republicans like Mr. Emmer say they are simply repeating Democrats’ own words, and they have been aided by Democratic stumbles. Ms. Ocasio-Cortez’s team published an early draft of the Green New Deal — which has been backed by several top-tier Democratic presidential candidates — that contained phrases those candidates did not endorse, including a call for economic security for “all who are unable or unwilling to work.” Ms. Ocasio-Cortez deleted the draft from her website, but Republicans took it and ran.

Representative Liz Cheney of Wyoming, the chairwoman of the House Republican Conference, called the plan — endorsed by 70 Democrats in the House and about a dozen in the Senate — “a fantasy” and insisted the erroneous draft be placed in the Congressional Record. Sean Hannity, the Fox News commentator, called it “a form of insanity.” The Republican National Committee issued a briefing paper headlined “The Democrats’ Burgeoning Love Affair With Socialism.”

In the Senate, where five Democrats are already running for president, Mitch McConnell, the majority leader, announced he would force a vote on the measure, drawing howls from Senator Chuck Schumer of New York, the Democratic leader, who slammed the move as a “cynical stunt” intended solely to put Democrats, including the presidential candidates, on the spot. A vote in favor of the Green New Deal could help them court progressives in the primary vote, but hurt in a general election.

“Democrats have handed Republicans this messaging on a silver platter,” said Andy Surabian, a Republican strategist and former Trump White House official, adding that he believed socialism and Democrats’ stance on abortion were more powerful lines of attack than anti-Semitism. “All these controversies dovetail perfectly with the president’s messaging at the State of the Union.”

Democrats dismissed the messaging as par for the course.

“We are not going to abandon socialist policies like Social Security — or is that ‘socialist security,’ is that what they call it?” Senator Richard J. Durbin of Illinois, the No. 2 Democrat in the Senate, said wryly. “The bottom line is the president is in campaign mode — maybe he’s always in campaign mode — and I just expect a lot of this to be thrown over the transom.”

Democrats say they intend to counter the offensive by talking about the issues they ran and won on: reducing health care costs and prescription drug prices, passing an infrastructure package and rooting out corruption in Washington.