I'd like to add, as well, that in two trips with a stroller we haven't had a problem at all. We have left the diaper bag stashed when we all went on Winnie the Pooh with no problems. Of course, I always take my purse/camera/videocam/etc with me. I was very surprised at how many purses and cameras I saw in stroller baskets in the various stroller parking areas. I'm not that brave (or naive) I guess! Just because it's Disney, doesn't mean it's 100% safe.
But we never had any problem with our strollers, or anything in them. I mean, who wants to swipe Pampers and wipes, anyway?