I'd go with the Beastie Boys (originally a punk band before switching to crossover), Faith No More (which mixed it with funk rock), and Red Hot Chili Peppers (also a funk rock band). And anything from
Judgement Night counts even if they were just temporary.
Also, Anthrax and Public Enemy collabed at times and did some legit crossover when they did.
And just to clarify why I'm no counting the previously mentioned bands, Blondie did only one song with a single rap section, Run DMC/Aerosmith's collab was just one song which jumped back and forth between styles, and ST were not really rapping but doing spoken word style lyrics side by side with sung lyrics. So they are precursors to the true crossover bands that came later.