Republican lawmakers in Arizona voted on Tuesday to put a ballot measure before voters in November that would make unlawfully crossing the border from Mexico a state crime. The proposal would give local police officers the power to arrest and jail unauthorized migrants, and would allow state judges to order deportations.

While immigration is the focus of campaigns across the country, the measure in Arizona is significant because it places the border crisis directly onto the ballot in a swing state that is seen as crucial in the presidential race.

Republicans are betting it will fire up anti-immigration conservatives and draw in otherwise unenthusiastic independents. And it could sit on a potentially crowded ballot, along with another measure protecting abortion rights, which Democrats hope will draw out more voters on their side.

The measure passed 31-29 along party lines. In floor speeches, Democrats called it an ineffective and racially biased measure that would break up immigrant families and hurt the state’s economy and reputation. Republicans focused on overdose deaths and migrants accused of murders, and called the ballot measure a necessary response to an unchecked “invasion.”

“The federal government has lost control,” said Representative John Gillette, a Republican. “We must act.”

Outside the Capitol, immigrant-rights supporters waved signs saying “No hate” and shouted their opposition in a last-ditch effort to pressure lawmakers to kill the measure, saying it would break up families and spread fear through immigrant communities.

But the viewing gallery inside the Arizona House was silent and empty. Republican leaders took the unusual step of closing it, citing the risk of disruption. Democratic lawmakers accused Republicans of hiding from the public.

If voters pass the border enforcement measure in November, it would mark a sharp U-turn for a state that has moderated its approach to immigration since the era when Sheriff Joe Arpaio of Maricopa County carried out raids and traffic stops that critics criticized as racial profiling.

In recent years, Arizona voters have approved in-state tuition rates for undocumented students and have rejected immigration hard-liners like Mr. Arpaio, who was ousted in 2016, and former President Donald J. Trump.

But Republicans say they believe voters are ready to embrace their new crackdown, because they are fed up with seeing thousands of migrants camped along the border wall and the growing death toll from fentanyl smuggled across the border.

Record numbers of migrant crossings have angered Democratic leaders and voters in cities like New York and Chicago, as well as Republicans, and have become a major re-election liability for President Biden. On Tuesday, Mr. Biden issued an executive order that would let him close the border when crossings surge.

Opponents say the Arizona ballot measure will do nothing to improve border security or prevent asylum seekers from arriving. Instead, they say, it will replicate the paranoia and turmoil that Latino and immigrant communities experienced after Gov. Jan Brewer signed S.B. 1070, a divisive state immigration-enforcement law passed by Republicans in 2010. That measure, which came to be known as the “show me your papers law,” sparked years of protests and litigation, and has since been partially struck down.

Arizona’s new ballot measure is similar to laws passed by Republicans in Texas and Iowa that challenged the federal government’s exclusive power to enforce immigration laws. The Biden administration has sued to block the Texas and Iowa laws, calling them unconstitutional.

Politicians on both sides said they expected the proposed Arizona law to face legal challenges if voters pass it. Gov. Katie Hobbs, a Democrat, has denounced the effort, but she does not have veto power to block Republicans from sending the measure directly to voters.

The measure, which would need a simple majority of voters to pass, is called the Secure the Border Act. Along with the policing provisions, it would also increase prison sentences for anyone who sells fentanyl that results in an overdose death, and would make it a state crime for undocumented workers to provide false information to the E-Verify screening system.

“This would certainly help draw Republican voters out,” said Mark Lamb, a cowboy-hatted sheriff from a conservative county south of Phoenix who is running in the Republican primary for the U.S. Senate. He had concerns about the measure’s price tag but said he would ultimately vote for it.

Democratic activists said the immigration measure could backfire by stirring up a wave of opposition from Latino voters and suburban moderates worried about the damage to immigrant families and Arizona’s reputation and economy.

Arizona’s population is 32 percent Latino, and many voters still have searing memories of S.B. 1070.

On a recent 100-degree Saturday, dozens of Latinos opposed to the measure gathered on the lawn outside the State Capitol to trade stories about how they had lived in fear and paranoia under S.B. 1070 and had seen immigration agents at their front doors. Shading their children under parasols and olive trees, they shouted the old farmworkers’ union slogan “Sí se puede” — “Yes we can” — and urged their neighbors to start organizing to register voters and defeat the ballot measure.

“We’re dead against it,” said Nieves Riedel, the mayor of San Luis, a small Arizona city that sits directly along the border wall. She said the city’s police force was already short 57 officers and could not handle the cost and time it would take to arrest hundreds of migrants. Sheriffs and prosecutors say local courts and jails would be overwhelmed.

“There’s only so much we can do,” Ms. Riedel said. “Our policemen and women are not federal agents. They’re not trained. What’s going to happen to our safety and security if they’re acting like Border Patrol agents?”

Mark Dannels, the sheriff of conservative Cochise County, has been one of Mr. Biden’s staunchest critics along the border, but he said the border security ballot measure would amount to little more than a giant new job for his officers without providing new funding.

“How the heck are we going to do this?” he said. “We don’t have the budget. We don’t have the resources.”

But conservative border ranchers like Fred Davis said Arizona had to do something. He often sees law enforcement officers chasing suspected human traffickers down the highway that bisects his ranch near Tombstone, and he regularly calls the Border Patrol to report migrants emerging from desert washes and thick brush near his property.

Republican lawmakers say the proposed law would allow the local sheriff to charge migrants like those with illegal entry — a misdemeanor for a first offense and a felony punishable by years in prison for anyone already convicted of entering illegally.

“Given the lack of control of the border, I just feel it’s something the state has to resolve,” said State Senator Ken Bennett, a Republican who voted for the measure.

He said the law was narrowly focused on border enforcement and would require the police to witness someone crossing the border or have a recording to make an arrest.

“That’s not stopping someone hundreds of miles inside the inner parts of the state,” Mr. Bennett said. “You got to see them with your own eyes or have technological evidence.”

But immigrant-rights activists said another line in the law allowing “any other constitutionally sufficient” probable cause would give law enforcement officers free rein to arrest unauthorized immigrants anywhere in Arizona.

Immigrant activists are already rallying against the measure but said they were worried it could easily pass in a state where many voters are upset about the surge in migrants.

Irayda Flores, a seafood importer in Phoenix who was born in Mexico, said she had spent years worried about losing her legal status as she fought to get permanent residency. Now, she said, she is disheartened that her immigrant employees or her son could face the same fears.

“We’re going back” to a worse time, she said. “The immigrant community, we pay taxes, we bring a lot to the table. And they treat us like criminals.”