Protests and arrests on college campuses exploded into the forefront of the presidential race this week, opening up a new line of attack for Republicans and forcing President Biden to directly address an issue that has divided the liberal wing of his party.

With Donald J. Trump largely stuck in a New York City courtroom for one of his criminal trials, Republicans have tried to use the protests as a political cudgel and a literal backdrop to attack Mr. Biden, casting him as weak and unable to…