WILMINGTON — Progressive challenger Kerri Evelyn Harris put U.S. Sen. Tom Carper on the defensive Monday as he continued a move to the left by embracing a $15 minimum wage and describing his 2006 vote to confirm a judge who is now a nominee for the U.S. Supreme Court as a mistake.

Sen. Carper, a Democrat seeking his fourth six-year term in the Senate, and Ms. Harris squared off in a debate at the Cab Calloway School of the Arts 10 days before the primary election. Ms. Harris painted the incumbent as out of touch and Sen. Carper cited votes he has taken and policies he has supported over the course of his 42 years in elected office.

It was the first time the two have debated.

Each candidate received loud applause throughout the event, although the audience seemed to favor Ms. Harris. The two Democrats went after President Trump on numerous occasions, blasting him as causing division, promoting policies that harm the environment and pushing “really stupid ideas on trade and on the economy,” as Sen. Carper termed it.

The candidates initially avoided direct attacks, but as the debate progressed, Ms. Harris began taking more shots at Sen. Carper’s record, criticizing him for supporting the Keystone Pipeline, taking donations from corporate political action committees and voting to confirm Brett Kavanaugh to the U.S. Court of Appeals for the District of Columbia Circuit

“All of your votes count, not just the one that you can pat yourself on the back for,” she said.

Sen. Carper admitted his 2006 vote for Mr. Kavanaugh, who has been nominated by President Trump to the Supreme Court, was one he would like to take back. He was one of four Democrats to back the nomination to the Court of Appeals.

“I voted my hopes over my fears. I will not do that again,” the senator said.

He urged the president to nominate to the nation’s top court Merrick Garland, the chief judge for the D.C. Circuit Court of Appeals who was nominated by President Barack Obama in 2016 but never received a Senate hearing.

Ms. Harris declared she could not support any pick from President Trump, noting he is currently under investigation for potential collusion with Russia in the 2016 election.

Both candidates called for more than doubling the $7.25 federal minimum wage to $15, with Sen. Carper urging Congress to index it to inflation so it automatically increases.

He has been skeptical about raising the wage floor above $10 in the past.

While Sen. Carper spoke out against the president’s immigration policy and pledged to continue defending the Affordable Care Act, Ms. Harris went further, calling for the abolition of Immigration and Customs Enforcement and for Medicare for all.

Most people have health care, she said, but many cannot afford to use it.

“Our premiums are too high. Our deductibles are too high. Our copays are too high,” she said.

“The problem is the cost of medication makes us decide whether to have to pay for food, pay for utilities or pay for prescriptions. People are literally dying because they only have access.”

Ms. Harris also took a position to the left of Sen. Carper’s on student debt, speaking in favor of “bailing out” college students and graduates. While saying it sounds nice, Sen. Carper called her position unrealistic and pointed to his support for increasing Pell Grants.

The candidates aligned somewhat on reducing the impact of big money in politics, although Ms. Harris once again went further than her opponent. She noted she has not taken any corporate donations and urged people to look at Sen. Carper’s campaign finance reports to see a contrast.

For his part, Sen. Carper cited the broad support he has received and spoke of his hope to end “dark money” donations by requiring all contributions to identify the source.

“For more years than I can count I’ve supported changing the way we finance our campaigns,” he said.

Challenged by Ms. Harris to back a bill that would provide $100 billion to combat the opioid crisis, Sen. Carper responded that the measure is merely an authorization, not a commitment, and he wanted to ensure he did not make an “empty promise” for money that could not be provided.

Throughout her campaign, Ms. Harris has characterized herself as a regular person who has at times struggled to get by, and she continued that Monday, arguing politicians too often put the interests of big business and the wealthy above those of the masses.

“We keep saying that if corporations do well, don’t worry, eventually it’ll get to you, but it’s not. What we’ve seen throughout history is when the middle class does well, corporations do well and the reverse has never, ever worked,” she said.

Each candidate received warm responses from the audience when they spoke out against President Trump, a Republican.

Asked how Congress should balance enforcing immigration law versus promoting fair and caring policies, Sen. Carper quipped, “The best thing we could do is change presidents.”

The president’s trade policies are harming workers, and his repeal of Obama-era climate rules are doing great damage to the planet, he said

“How stupid can we be?” he asked in reference to a proposal to lower car emission standards.

Asked what the best thing the president has done thus far, Sen. Carper credited him for uniting a “badly fractured” Democratic Party and Ms. Harris said he has caused many more people to get engaged in politics.

She said President Trump has attempted to weaken the working class, sowing division among Americans.

“He’s done it by making us point fingers at each instead of realizing that he passed a tax code that is destroying us every single day it stays active,” she said.

Predictably, both campaigns claimed victory in email blasts sent out shortly after after the debate.

The Democratic debate was preceded by Republican candidate Gene Truono answering questions from a panel on a variety of subjects. Mr. Truono’s opponent in the primary, Rob Arlett, was unable to attend due to a scheduling conflict, according to the Arlett campaign.

Mr. Truono’s positions were diametrically opposed to most of the ones taken by Sen. Carper and Ms. Harris, as he defended the White House’s policies. Mr. Truono attacked Mr. Arlett as well, claiming he has consistently voted in favor of land developers during his time on Sussex County Council.

The primary will be on Thursday, Sept. 6.