I bought these two books:

The Art of Contemporary Travis Picking [With CD] by Arnie Berle (Bk&Cass) and Beginning Fingerstyle Blues Guitar (Guitar Books) by Mark Hanson. They're both really good books that everyone recommends when someone wants to learn fingerstyle guitar

With which one should I begin?
First of all: wow....that is one hell of a bump from the original poster. Crazy that nobody answered that one!

I've had bad experiences learning to "play guitar" from books. Except for musical theory (if they mention musical theory at all) they are worthless because it comes down to muscle memory anyway. How do you get muscle memory? Not from reading books...you only get that by playing. Maybe you are different than I am, but I got nothing from books when it came to learning guitar, except for learning chord diagrams in the beginning.

I'd highly recommend you use the free lessons available on youtube....there are tons of good fingerstyle lessons online elsewhere as well. I think a person absorbs more information if they can hear/watch the instructor, rather than just reading something in a book.

Good luck. For what it's worth...Travis Picking was giving me problems (self taught here) until I got a lesson about 6 months ago. The instructor showed me what I was doing wrong, literally, in 5 minutes. Do yourself a favor and find a good, enthusiastic instructor.
I think the problem is that you mentioned two specific alternatives, and no-one has an opinion between those two. I would likely take a punt on Mark Hanson's stuff, he is well known and has been doing (did?) instructional articles in AG Mag for a long time.

Like TR says, it might be a good idea to have some one show you the principles of alternating bass fingerpicking. The basics are very simple, but it takes a lot of practice to get into "autopilot" mode.
If I were to suggest a book I would recommend Guitar Principles by Jamie Andreas. It will teach you proper technique. From there you can go on to learn any type of fingerstyle you'd like. I hit a wall when I was learning fingerstyle, that I later figured out was due to bad technique. Jamie's book directed me towards proper technique which allowed me to improve much more quickly - nowadays I apply her rules of motion with whatever new technique I'm learning. That and learning to count out loud while playing (so to improve my rhythm drastically!) have been two of the most useful tools for me in progressing my fingerstyle.