Can't you fence them out?

I've heard it will work quite well. If you have it above the ground, it may make weeding difficult. Otherwise if it is flat on the ground it should discourage the cats. There are a few drawbacks, such as getting large root crops through the wire when you pull them, but otherwise you should be fine. It wouldn't hurt to have a good sandbox nearby to help divert the cats, just to be safe.

Depending on how large your beds are, you could use 2"X4" welded wire fencing, clip and bend the corners(it bends easily) making it like the top of a gift box(and however high you need it). The spaces are big enough that your cats won't get their feet caught, they won't be able to use it for a litter box, and you can just lift it off to work your soil. Allowing the cats to have their own small raised bed or sandbox as Beeone suggested is a good idea.

How about using plastic mulch or landscape fabric?

That would give you other benefits such as weed control, moisture retention, etc., as well as foiling the cats.

Jim

That might work best of all. I would be able to cut the fabric if I needed to and tape or cover it over with more fabric! I have a light fencing around it right now that I have to unhook and rearrange or bend over every time I have to get in there.
In fact, right now I have a mini hoop house with pvc hoops and plastic. The momma cat was caught staring dreamily at the warmth inside...