Harvard and others have a choice — they can provide the information requested by DHS or not be allowed to bring foreign students onto campus. Foreign students need a visa, and those are granted by the current administration.
There are a number of smaller colleges that have half their student body from other countries because those typically pay full tuition (meaning that they pay for three people). If the colleges lost that revenue they might have to cut worthless victims studies departments or layers of useless administrators that never set foot in a classroom.
The horror!