I wouldn't sweat that one. If the 64-bit processor is one or two dollars more (literally), then I'd go for it. Otherwise, no big deal. Linux works great on both 32-bit and 64-bit platforms.
The reason for a 64-bit processor is simple: memory addressability. If you plan to stuff your system with more than 4 GB of memory, then it's worth considering. If you never plan to do that, don't bother.
Speaking of which, that's the single biggest factor in causing a computer to become obsolete. Buy a system that lets you stuff lots of RAM in it, and upgrade the RAM first. Don't worry about the processor. One processor might be 30% faster than another, but RAM is multiple factors of 10 faster than a hard disk.
Linux is also very good at extending the life of your computer equipment. I've got a ten year old computer that runs just fine with Linux for web browsing, e-mail, networking, etc.
Finally, I'm a big fan of
Knoppix. Knoppix is a Linux distribution that's contained entirely on a single CD-ROM, and it boots from the CD. So you can run Linux without installing Linux on the hard disk. (It won't touch your PC's hard disk unless you want it to.) That's a great way to try out Linux if you're concerned about doing something to your PC.
Otherwise there are some great Linux distributions out there for the beginner, such as Lycoris, Lindows, and Xandros.
I'm running Linux on my IBM ThinkPad, and I love it. I encourage everyone to at least try Knoppix and start the process of getting familiar with it and switching over. You'll be so glad you got off the Microsoft nickel-and-dime-you-to-death treadmill.