Top colleges have pledged to become more socioeconomically diverse, but the admissions edge many give to children of alumni may make that goal harder to achieve.

At the University of Notre Dame, the University of Virginia and Georgetown University, the admission rate for legacies is about double the rate for the overall applicant pool, according to data from the schools. At Princeton University, legacies are admitted at four times the general rate, or roughly 30% compared with about 7% overall over the past five years, the...