WASHINGTON—President Trump announced a 30-day ban on some travel from Europe into the U.S. and said he would act to offer financial assistance to those affected by the coronavirus pandemic rapidly spreading across the country and around the world.

Mr. Trump, in a national address from the Oval Office on Wednesday, said the travel suspension would begin Friday at 11:59 p.m. and would exclude the U.K. He suggested the restrictions would also apply to cargo, but the White House later said they applied only to people and declined...