The Brussels terrorist attacks should push the U.S. to tighten security at airports, increase screening of aviation workers and pump more federal money into transit, Sen. Charles E. Schumer said Tuesday — though he rejected GOP calls to tighten immigration and visitor visas for Muslims.

While other Democratic leaders offered general words of support for Belgium Tuesday, Mr. Schumer, a New York Democrat who is poised to become Senate Democrats’ leader next year, laid out several specific policies his party can get behind as Americans search for ways to try to prevent similar attacks here.

“I believe that the administration’s doing a very good job fighting terrorism, but you can never be too careful,” he said.

He said airport screeners need better training to focus on terrorist threats, and said airport workers themselves need to be vetted better to make sure terrorists aren’t able to sneak into sensitive places by infiltrating that workforce. He said an upcoming debate over the Federal Aviation Administration will give Congress a chance to have that debate.

Thirty people were reported dead and more than 200 were injured in several bombings Tuesday morning in Brussels. The Islamic State has claimed responsibility for the attacks, which came just days after authorities in Belgium apprehended the man accused of orchestrating last year’s Paris terrorist attack.

After the latest bombings, Democratic presidential candidates called for solidarity with Europe, while Republican presidential hopefuls Donald Trump and Sen. Ted Cruz laid out specific plans they said would help keep the U.S. safe.

Mr. Trump reiterated his push for a temporary ban on admitting Muslims to the U.S., and said the country should consider halting the visa waiver program that allows many Europeans to enter the U.S. without facing an interview or in-person screening. Foreign fighters who trained with the Islamic State but who hold European passports could use the waiver program to gain a foothold in the U.S., Republicans say.

Mr. Cruz said the bombings should force President Obama to revoke his plans to bring 10,000 Syrian refugees into the U.S. this year.

Mr. Schumer rejected a ban on Muslim entry or stiffer monitoring of Muslims already in the U.S.

“That’s not what we do in America. We don’t take a race, religion, creed, color and say we’re monitoring everybody. We look for terrorism where it is, and we pursue it relentlessly,” he said. “That’s what we should do, that’s what I believe the administration is doing.”

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