Democrats in battleground states are growing increasingly anxious about President Biden’s low approval ratings, worrying that voters’ persistent antipathy toward his leadership could not only cost the party the White House but also weigh down the candidates who are sharing the ballot with him.

These Democrats fear that the Biden campaign is late in building a strong organization in the handful of states that are likely to determine next year’s presidential election. They point to polling numbers showing Mr. Biden lagging far behind Democratic candidates for Congress in those states and struggling among key groups of voters, including Black and Latino Americans.

In Arizona, Democratic polling has found Mr. Biden losing Hispanic voters to former President Donald J. Trump in Maricopa County, which includes Phoenix and represents more than 60 percent of voters in the state. In Michigan, where Mr. Biden’s approval rating is a striking 15 percentage points behind that of Gov. Gretchen Whitmer, a fellow Democrat, he has lost ground with Black and Arab American voters. And in Georgia, officials say the Biden economic message has not broken through to voters, in part because voters have seen Gov. Brian Kemp, a Republican, take credit for many of the new projects in the state.

“I’m extremely concerned,” said Mayor Van Johnson of Savannah, Ga. “President Biden is a man of great character. Certainly, he’s a president of great accomplishments. But that is not translating to southeast Georgia.”

Mr. Biden’s top aides and most fervent surrogates have for months insisted that the race will change once voters understand that Mr. Trump will be the presumptive Republican nominee, potentially as soon as next month. At that point, the Biden team argues, the campaign will transform from a referendum on Mr. Biden to a choice between the president and Mr. Trump, whose brand of right-wing Republicanism has lost most major elections since he won the 2016 election.

Mr. Trump has been indicted on 91 felony charges and has, at least for now, been barred from the ballot in Colorado. The former president is scheduled to sit for the first of his four criminal trials in March, though that could be delayed. While those events have bolstered his appeal among Republican primary voters, the Biden team believes they will turn off independent voters.

Mayor Johnson, among others, is not convinced. He called the choice argument “a passive strategy” and said Republicans were far more excited about next year’s election than Democrats.

“I don’t see any passion, any excitement, nothing,” he said. “It might be a situation of too little, too late.”

The good news for Mr. Biden is that his party is on a winning streak that dates to when Mr. Trump took office in 2017. Since then, Democrats have won election after election, including key governor’s and Senate races, and they outperformed polling and historical trends in last year’s midterms.

The Biden operation has in recent weeks finally announced staff hires in three general election battleground states. Key economic indicators are improving, even though voters have not yet given Mr. Biden credit for that progress.

Some signs suggest the campaign is pivoting to a more assertive messaging strategy. When Mr. Trump late last month said he would seek to repeal the Affordable Care Act, officials working on battleground Senate campaigns noted in their group chats that it was the first time that the Biden campaign had come to them proactively with requests to amplify the president’s message.

On Thursday, Julie Chávez Rodríguez, Mr. Biden’s campaign manager, released a memo titled “Why Joe Biden Will Win in 2024.” In it, she said that there would be “thousands of staff dedicated to Team Biden-Harris” by early summer, and that, in the new year, Mr. Biden and Vice President Kamala Harris would devote more energy to drawing a contrast between themselves and Mr. Trump.

“There’s been no lack of coverage on polls about Joe Biden,” Ms. Chávez Rodríguez wrote. “But it’s important to remember Donald Trump, extreme MAGA Republicans and their dangerous ideas are extremely unpopular.”

Yet, Democratic officials and strategists acknowledge it will be harder for their party’s candidates to outrun Mr. Biden in 2024 when he is leading their ticket. When Democratic state party chairmen gathered in Washington last week, some shared a troubling new trend they had been hearing: Some young voters are blaming Mr. Biden for the Supreme Court’s Dobbs decision ending the constitutional right to abortion because he was the president when the court made its decision.

Representative Debbie Dingell of Michigan, who was one of the first Democrats to warn that Hillary Clinton was in danger of losing to Mr. Trump in 2016, said she had seen similar overconfidence in the Biden team.

“In 2015 and 2016, I warned people and nobody believed me,” Ms. Dingell said. “He’s got to do more than just draw the contrast.”

Dan Sena, a former executive director of the House Democrats’ campaign arm, said Democrats running for congressional offices should try to make their races as much about local issues as possible, a well-worn strategy for candidates running with an unpopular president of their own party.

“The presidential race speaks in such broad terms,” he said, “that it really affords Senate and congressional candidates the opportunity to localize their races, which creates a natural space between them and the president.”

Some Democratic candidates in tough races are already adopting the approach.

Representative Colin Allred, a Dallas-area Democrat challenging Senator Ted Cruz, said he was focused solely on defeating his opponent, not on Mr. Biden and the presidential campaign.

“I really believe that we can’t afford six more years of Ted Cruz. It’s an election that’s going to outlast the next presidential year,” he said. “I’ll let the president run his campaign.”

And in Michigan, Representative Elissa Slotkin has privately told allies that Mr. Biden’s low standing will hurt her in her Senate race next year, according to two people familiar with the conversations. Ms. Slotkin’s campaign spokesman, Austin Cook, said she “looks forward to running with President Biden” next year.

Mr. Sena also noted that the dynamics could be different this year, given the number of independent candidates mounting presidential bids. While those candidates could claw some votes away from Mr. Biden, their supporters may not have a well-known alternative to the two parties in congressional and state races.

“There’s no doubt it will be very hard to outrun the president in 2024,” he said, “but the question is whether the ballot dynamics are such that the independent candidates afford opportunity to over perform the president.”

Morgan Jackson, a Democratic consultant in North Carolina who is working on the campaigns for governor and attorney general there, said he is a believer in the Biden campaign’s mantra that voters have not yet tuned into the 2024 race.

“What you’re seeing from voters is they are not that engaged, they don’t really like their options and they don’t know what they’re going to do,” Mr. Jackson said.

The Biden campaign has announced staff hires in Michigan, Nevada and Wisconsin, as well as South Carolina, which is home to the first primary recognized by the Democratic National Committee and is serving as a test of Black voters’ enthusiasm for the president.

Mr. Biden has repeatedly questioned whether polling in battleground states accurately reflects the reality of the race. Indeed, a New York Times/Siena College poll released this week that showed Americans disapprove of Mr. Biden’s handling of Israel’s war against Hamas also found Mr. Biden effectively tied with Mr. Trump among registered voters. The results echoed the findings of a series of other national surveys over the past two weeks, which showed Mr. Biden with a slight lead in a tight race.

When reporters asked Mr. Biden, who had stopped by his campaign headquarters on Sunday for a visit with staff, why he was losing to Mr. Trump in polls, he replied that the news media were reading “the wrong polls.”

Kevin Munoz, a Biden campaign spokesman, said the campaign was “focused on the end game: winning next November.”

Some Democrats say that the campaign is making a mistake by waiting to fully engage. The campaign, said Simon Rosenberg, a veteran Democratic strategist, needs to start engaging the part of its coalition that has drifted.

“He’s underperforming, but we know that once the campaign ads start and the campaign turns on, a big chunk of our wandering coalition will come back immediately,” Mr. Rosenberg said.

And, Mr. Rosenberg says, a more active campaign could help with a perpetual issue for Democrats: nerves.

“There is anxiety in the family right now, and the best way to deal with it is to give everyone something to do,” he said.