By Amanda Becker

(Reuters) - Democratic presidential front-runner Hillary Clinton said on Wednesday the attack in Brussels that killed more than 30 people is the "latest brutal reminder" that more must be done to defeat Islamic State militants, including by European Union member countries.

In an address at Stanford University in California, Clinton said the United States and Europe should take a "harder look" at protocols at airports and other "soft sites" that are outside security perimeters.

Clinton also said "there is much we can do to support our European partners" but "there is also more they can do to share the burden with us."

Clinton said she would like to see more European countries investing in defense and security in the way Germany has during the Obama administration.

"The most urgent task is stopping the flow of foreign fighters to and from the Middle East" who are citizens of France, Germany, Belgium and the United Kingdom with European passports that make it easier for them to cross into Syria and return radicalized, Clinton said.

Clinton said that many European nations do not currently alert their neighbors when they turn away a "suspected jihadist" at their border or when a citizen's passport is stolen. European Union countries also need to share traveler information more readily, Clinton said.

"It's actually easier for the United States to get flight manifests from EU nations than it is for EU nations to get them from their own neighbors, thanks to an agreement that the United States negotiated when I was secretary of state," Clinton said.

Additional steps that could be taken in Europe are the creation of a "new, unified, European border and coast guard" to strengthen the external borders of a continent that is under "unprecedented pressure from refugees and migrants," Clinton said.

Clinton also praised past partnerships between the United States and Europe, calling the NATO alliance between North American and European countries one of the best investments that America has made.

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Republican front-runner Donald Trump has in recent days said that the United States should rethink its involvement in the decades-old alliance.

Clinton also took aim at Trump, along with presidential rival U.S. Senator Ted Cruz of Texas, when she slammed "offensive, inflammatory rhetoric that demonizes all Muslims," including those who could be partners in the fight against terrorism.

(Reporting by Amanda Becker in Washington; Editing by Eric Beech and Andrew Hay)