If you don't finish both sides the wood will be more subject to warping. That said, it is a common practice with built-ins and isn't typically a problem, especially if you have real BB and not a substitute. If mold and rotting is a concern you have much bigger problems than whether your book cases are finished-- patch the hole in the roof!!

Since you are DIYing it just finish the backs this afternoon and then there won't be an issue. A couple coats of water-based finish appied an hour or two apart will do the trick, you can then stain and finish the show surfaces at your leisure.

For myself, I'd fully finish them in the shop prior to installation. Much less hassle if, for example, you're spraying dye stain (something I'd highly recommend on BB to avoid a blotchy surface).

Please be sure to make sample boards, using leftovers from the actual wood used in the book cases if at all possible, following the full sanding and finishing schedule for the show surfaces on the samples. The number of bad surprises possible with applying dyes and stains continues to amaze me. Please don't let anyone talk you into applying a pigment type stain directly to raw wood; it seldom ends well. Test extensively on several different pieces, including your edge banding if you are using it, if you go that route.

Get either Bob Flexner's or Jeff Jewitt's books on wood finishing if you haven't done this before.

Haha, nothing as bad as holes in the roof, just my rudimentary knowledge of woodworking that tells me wood should be protected on all surfaces. This is lots of great information. I've done a bit of staining and finishing over the years. But this piece is a significant investment, so we want to make sure it looks nice. Thanks for the advice. It looks like I have a bit of research to do. Fortunately we have some time to get the staining/finishing done.

Install them and move on. If you want to protect the backs roll on a coat of water based poly. Focus your energy on the part that shows.