Hillary Clinton is accusing Donald Trump of giving “aid and comfort” to Islamic terrorists, declaring his anti-Muslim rhetoric helps the Islamic State group and other militants recruit new fighters. Mr. Trump is insisting the U.S. should “use whatever lawful methods are available” to get information from the Afghan immigrant arrested in this weekend’s bombings.

As Mr. Trump supporters at a packed rally in Florida shouted “Hang him!” the Republican presidential candidate mocked the fact that Ahmad Khan Rahami, a 28-year-old U.S. citizen originally from Afghanistan, would receive quality medical care and legal representation.

“We must deliver a just and very harsh punishment to these people,” he said. “These are enemies, these are combatants and we have to be tough, we have to be strong.”

Both candidates moved swiftly to capitalize on investigations into a weekend of violent attacks bombings in New York and New Jersey and stabbings at a Minnesota mall casting themselves as most qualified to combat terrorism at home and abroad.

Ms. Clinton touted her national security credentials at a hastily arranged news conference outside her campaign plane, accusing Trump of using the incidents to make “some kind of demagogic point.”

Ms. Clinton and her team see her experience and what they say is her steady judgment as key selling points for her candidacy. On the campaign trail, she frequently invokes her role in the 2011 raid that killed Osama bin Laden, describing to voters the tense atmosphere in the White House alongside President Barack Obama at that moment.

But while much of the foreign policy establishment has rallied around Ms. Clinton, Mr. Trump’s anti-immigrant rhetoric, promises to close U.S. borders and vows to aggressively profile potential terrorists have fueled his presidential bid.

On Monday, he called for tougher policing, including profiling foreigners who look like they could have connections to terrorism or certain Middle Eastern nations.

“This isn’t just a matter of terrorism, this is also really a question of quality of life,” he said. “We want to make sure we’re only admitting people into our country who love our country.”

Pointing to her Monday morning comment that Mr. Trump’s words give “aid and comfort” to Islamic extremists, his campaign said Ms. Clinton was accusing him of treason, going beyond the bounds of acceptable campaigning and trying to change the subject from her own failures.

She insinuated that Islamic militants, particularly those affiliated with ISIS, are rooting for Mr. Trump to win the White House. She said, “We’re going after the bad guys and we’re going to get them, but we’re not going to go after an entire religion.”

Mr. Trump agreed terrorists have a preference- They “want her so badly to be our president.”

Ms. Clinton met late Monday with the leaders of Egypt, Ukraine and Japan in New York City, where they gathered for the United Nations General Assembly. Mr. Trump met with Egyptian President Abdel-Fattah el-Sissi.

Mr. Trump has tried to appear more statesmanlike as the November election approaches. Still, he suggested it’s fine if some world leaders feel uneasy about him.

“Well, maybe that’s a good thing, not a bad thing. Right now the world has no respect for our country, they have no respect for our president, whatsoever,” he said in an interview on Fox News.