WASHINGTON — A draft Supreme Court opinion that would overturn Roe v. Wade has jolted the battle for control of state legislatures, where the next stage of the struggle over abortion rights is likely to play out.

If the draft opinion that was leaked is not substantially changed and Roe is in fact overturned, about half of U.S. states are likely to ban or sharply limit abortion, according to a New York Times analysis. But in what otherwise looks to be a difficult year for Democrats, party strategists see the looming rollback of reproductive rights as an opportunity to galvanize key voting blocs, limit Republican gains and perhaps even pick up seats in certain states.

“We don’t know exactly what the political environment will be,” said Jessica Post, the president of the Democratic Legislative Campaign Committee, which helps Democratic candidates for state legislature. “But abortion has the potential to be a game-changing issue.”

State legislative races are not glamorous, high-dollar affairs. But the Democratic group had its biggest fund-raising day of the year after the publication of Justice Samuel A. Alito Jr.’s draft opinion, and raised more than $650,000 over 48 hours.

The surge reflected a growing recognition among Democratic donors and voters, Ms. Post said, that “the federal government isn’t coming to save us.”

In a new memo and accompanying website shared with The New York Times, the committee outlined its strategy for the 2022 midterm elections. The memo acknowledges how, in the 2010 election cycle, “Democrats were caught flat-footed at the state legislative level of the ballot and our chambers took a hard hit.”

This year, the memo reads, “We will not be caught off guard.” The group has already raised $30 million, and expects to raise $50 million to $60 million by Election Day.

Democrats plan to concentrate their energies in two main areas. They are defending their majorities in Colorado, Maine, Nevada, New Mexico and Minnesota, where they control statehouses. And they hope to flip legislatures in Michigan, New Hampshire and Minnesota, where Republicans have slim Senate majorities.

Democrats also see a somewhat slimmer chance to erode what they call Republicans’ “structural advantage” in Arizona, Pennsylvania and the Georgia House. The redistricting process in Arizona, led by a nonpartisan commission, produced new maps that still give Republicans an edge despite demographic shifts in the state that favor Democrats. And in Pennsylvania, Republicans’ majorities are large enough that it would be difficult for Democrats to overtake them even in a more favorable national environment.

“We know that this is a long game,” Ms. Post said. “Our goal is to slowly chip away at Republican power in the states.”

The memo’s cautious tone reflects the defensive crouch Democrats find themselves in amid soaring inflation and doggedly low approval ratings for President Biden — two confounding factors strategists often refer to euphemistically as “the environment.” One of the party’s greatest fears is that many of the voters who turned out for Mr. Biden in 2020, frustrated by his performance in office, will stay home in 2022.

“Even the D.L.C.C. is now admitting that Joe Biden is dragging down state Democrats and that they are in huge trouble ahead of November. It turns out mimicking the president’s radical agenda in states like Arizona and Pennsylvania is a losing strategy, which is why the D.L.C.C. is apparently waving the white flag in those battlegrounds despite vowing to flip them earlier in the cycle,” said Andrew Romeo, communications director for the Republican State Leadership Committee.

Justice Alito’s leaked draft “has the potential to be a watershed moment in a midterm cycle where Democrats face a historically difficult political environment and defeat looms large in tough races,” wrote Molly Murphy, a Democratic pollster, in a memo shared by a Democratic colleague.

The memo urges Democrats to tell voters that Republicans are pushing to take away an existing right, while cautioning against “overreach” by seeking to change the status quo on abortion law.

“The Supreme Court decision means that each state will now be allowed to criminalize abortion and ban it even in cases of rape, incest and life of the mother,” Ms. Murphy wrote.

Other indicators of abortion’s potential impact on state legislative races are more anecdotal in nature. Amanda Litman, a co-founder of Run for Something, which recruits young progressives to run for office, said her group had seen “a meaningful spike in candidate recruitment” since Justice Alito’s draft leaked.

“I expect this is only the beginning,” Ms. Litman added.

Gaby Goldstein, a co-founder of Sister District, a progressive group that backs Democrats in state legislative races, predicted that the “vitriol” and sweeping scope of the draft opinion could also enlist other communities, such as L.G.B.T.Q. voters, to embrace the cause of reproductive rights as their own.

Democrats are targeting state legislative races in states with major governor’s races, hoping to piggyback on the turnout and energy from the top.

Their best chance at a pickup could be in Michigan, where Gov. Gretchen Whitmer is squaring off against a boisterous field of Republicans. On Wednesday, the Democratic Governors Association said it was investing $23 million to bolster Ms. Whitmer’s re-election bid.

Ms. Whitmer has leaned into the abortion-rights fight, including by filing a lawsuit demanding that the State Supreme Court clarify “whether Michigan’s State Constitution protects the right to abortion.” A 1931 law barring abortion is poised to go into effect if Roe is overturned, and the Republican majorities in the State House and Senate have no intention of stopping it.

To varying degrees, Ms. Whitmer’s prospective Republican opponents back the abortion ban. One of the candidates, Garrett Soldano, a chiropractor, has made national headlines by urging rape victims to carry their pregnancies to term.

“God put them in this moment,” he said in an interview for the podcast “Face the Facts With April Moss.” He continued: “And they don’t know that little baby inside them may be the next president, may be the next person that changes humanity.”

National liberal groups, including Planned Parenthood and the American Civil Liberties Union, are pouring money into a ballot initiative that would keep abortion legal in Michigan.

New maps, produced by a nonpartisan redistricting commission, have made the Michigan State Senate an especially tempting target for Democrats. Thanks in part to aggressive gerrymandering, Republicans have controlled the body since the early 1980s. They currently hold a six-seat majority.

Jim Ananich, the top Democrat in the State Senate, said the chamber was “very ripe for picking up,” pointing to the new maps and shifting voting patterns among college-educated suburban women. Republicans’ unyielding positions on abortion, he said, would allow Democrats to portray them as out of step with most Michigan voters.

“The public is not looking for a radical agenda,” Mr. Ananich said. “They just want us to focus on them.”

Republicans find themselves torn between dueling imperatives: their base’s enthusiasm for cultural crusades, such as banning abortion, versus ordinary voters’ concerns about paying for gasoline and groceries. The environment is so favorable, one Republican consultant said only half in jest, that he would advise candidates to “take a long vacation and come back in November.”

Chaz Nuttycombe, an election forecaster, has calculated that Republicans stand to pick up more than 100 state legislative seats in November. Winning any new chambers would be an “uphill climb” for Democrats this year, he said.

Michael Behm, a lobbyist who specializes in state legislatures, agrees with that sentiment. “They’ve got some serious headwinds in front of them” he said, “caused by many things that are out of their control.”

Mr. Behm confessed that he was not yet sure which political party the spotlight on abortion would ultimately help, but said the end of Roe could certainly “shake everything up.”