I know about the situation at St. Paul's, and it was indeed horrific. But what about all the elite day schools that have had similar scandals, like St. Ann's, Horace Mann, and Poly Prep in NYC alone? Also, if I remember correctly, many of the St. Paul's abusers were students, not teachers.
For me, public vs. private is a more meaningful distinction than boarding vs. day. Private schools have more leeway in the people they hire; I doubt Winston Nguyen would have found a job at a public school with his criminal record, but St. Ann's was happy to have him.