Yes, there are bike racks outside pretty much every campus building—biking is very popular. The bike lane on University is very clearly defined and wide (this would probably be the main road your son would take from Witte to his classrooms). In general I’ve always felt safe biking on campus, and bringing my bike was one of the best decisions I’ve made transportation wise. From my experience, bike theft is not a big concern on campus.
No one rides a bike up Bascom Hill because it’s pedestrian only, and if he has a class in Bascom hall, I recommend parking his bike between the Physics building and the botanical garden and walking the side path up Bascom. Otherwise, the biggest hill you’d have to go up is Charter St, but there is a midway point where it intersects with Linden Dr. with lots of bike racks on both sides of the street and easy access to any building on Charter.
If you don’t want the hassle of transporting the bike back home over the winter, Ogg residence hall (a block away from Witte) has a covered bike parking lot—there are no bike parking permits or stickers, so as long as you find a spot, you can keep your bike there over the winter.