Nearly seven months after the Israel-Hamas war began, the demonstrations convulsing college campuses nationwide are exposing fresh tensions within the Democratic Party over how to balance free speech protections and support for Gazans with concerns that some Jewish Americans are raising about antisemitism.

From New York and Los Angeles to Atlanta and Austin, a surge in student activism has manifested in protest encampments and other demonstrations, drawing significant police crackdowns and sometimes appearing to attract outside agitators. The protests also have emerged as the latest flashpoint in the internal Democratic debate over the war.

As scenes of campus turmoil play out across the country in the final days of the school year, the moment also carries political risk for a party that has harnessed promises of stability and normalcy to win critical recent elections, and faces a challenging battle for control of the government in the fall.

“The real question is, can the Democrats again portray themselves as the steady hand at the helm?” said Dan Sena, a veteran Democratic strategist. “Things that create national chaos like this make that harder to do.”

Mr. Sena and other Democrats have argued that Americans have good reason to associate their opponents with chaos: Former President Donald J. Trump faces multiple criminal cases; the narrow, fractious House Republican majority has its own divisions concerning Israel and free speech; some Republicans have urged National Guard deployments to college campuses; and for years, Republicans have faced criticism over antisemitism in their own ranks.

But since the Hamas-led attack on Israel on Oct. 7, and the Israeli military response that has killed more than 30,000 people, according to local authorities, the fight over American policy toward Israel has been especially pronounced on the left.

Most Democrats say they both support free speech and condemn antisemitism, and consider criticism of the Israeli government to be fair game. But in seeking to address an intractable conflict marked by competing historical narratives, debates over how to distinguish between legitimate criticism of Israel and antisemitic speech are fraught and reaching a fever pitch on campus.

To some lawmakers who have visited encampments and attended demonstrations, the students are part of a long tradition of campus activism, and their free speech rights are at risk. Incidents of antisemitism, they say, do not reflect a broader movement that includes many young progressive Jews.

Representative Greg Casar of Texas went to the University of Texas to show solidarity with demonstrators, linking their activism to that of students who opposed the Vietnam and Iraq wars.

“So often, history ends up vindicating those who call for peace early,” he said. “I do think that more and more members of Congress will start to show up at these events and start to hear out more and more of where the students are coming from.”

Asked about instances in which demonstrators around the country have used antisemitic language, Mr. Casar replied, “those people suck.”

“They’re not a part of the peace movement,” he said. “Anybody that’s motivated by hatred — be it racism, Islamophobia, antisemitism, hatred of any form — they’re not peaceful.”

But to other Democrats, instances of intimidation and harassment described by some Jewish students are a defining feature of the campus movement.

Nowhere have those tensions been more clear than at Columbia University, which has become both an epicenter of the protest movement and a focal point for its detractors.

Democrats including President Biden, House and Senate leaders and prominent Senate candidates such as Representatives Adam Schiff in California and Ruben Gallego in Arizona have condemned antisemitic harassment around Columbia.

Other Democrats have sought to show solidarity in person with Jewish students who have described feeling unsafe. Representative Jared Moskowitz, a Florida Democrat, recently visited the campus with several other Jewish lawmakers.

Some in his party, he said, were downplaying the hard-line nature of some of the demonstrations.

“There are people who are peaceful, and there are not,” he said. “But there’s a denial from my friends on the left,” a view that “‘everyone’s peaceful, there’s no antisemitism.’”

He declined to name names, though he and Representative Alexandria Ocasio-Cortez have sparred on social media. Ms. Ocasio-Cortez of New York, one of several progressive lawmakers who have visited the Columbia encampment, has also condemned “horrific people wandering outside” Columbia’s campus who espouse “virulent antisemitism.”

But broadly, Mr. Moskowitz argued, some on the left who rightfully criticized antisemitic chants from “white, Aryan-looking men with tiki torches” rallying in Charlottesville, Va., in 2017 seemed reluctant to denounce threatening speech when it came from liberal-leaning Americans.

“I don’t see the same level of outrage,” Mr. Moskowitz said. “It’s politically inconvenient now.”

Representative Jerrold Nadler of New York, a long-serving Jewish member of Congress, has also expressed concerns about antisemitism. But he said his party was consistent in calling out bigotry, in contrast to many Republicans, pointing to Charlottesville. (Mr. Moskowitz shared that assessment about Republicans.)

“Democrats are willing to call out antisemitism, wherever it is, and certainly there’s been some antisemitism on campuses,” Mr. Nadler said, though he questioned how representative the demonstrations were of the student body.

Lauren Hitt, a spokeswoman for Mr. Biden’s campaign, said that “while Donald Trump stood proudly with white supremacists and encouraged violent crackdowns on peaceful demonstrators,” Mr. Biden defends the First Amendment and has “strengthened protections against antisemitism and Islamophobia.”

In Georgia, where demonstrators at Emory University were subdued forcefully, State Representative Ruwa Romman said that “there is no room for antisemitism in this movement.”

But she warned against focusing on a “few agitators” over the “thousands of students who are welcoming, who believe in a multiracial, multicultural, multi-faith world.”

“When we lose young people, we’re not just losing at the ballot box,” said Ms. Romman, a Democrat who is Palestinian. “We’re losing them in the entire electoral apparatus.”

In the meantime, some Republicans are seeking to paint the whole Democratic Party as extreme and overly attuned to concerns of Ivy League protesters.

Democrats “are demonstrating that they’re listening to a very small, very radical, very online segment of their base that is not representative of the broader electorate,” said Jack Pandol, a spokesman for the House Republican campaign arm, which is selling T-shirts that allude to a profanity aimed at Hamas.

Former Representative Steve Israel, who led the House Democratic campaign arm, said that while Republicans might see a messaging opportunity, it was far too early to determine whether it would be potent come November.

“Campuses generally clear out in summer, the energy on this issue may dissipate and the question will be whether it returns in the fall,” he said. “The answer to that isn’t here. It’s in the Middle East.”