Crime does exist at WDW. I'm glad to browse thru some of these posts to see that Housekeeping doesn't seem to be an issue, but in public there's a good amount.
Disney just does a great to excellent job of preserving its persona to its guests.
Most of the crime is against Disney itself, as opposed to guest-on-guest offenses.
The Magic Kingdom has a County Sheriffs office, complete with a holding cell, above Uptown Jewelers on Main St.
Counterfeit Travelers Checks and Currency along with regular theft in Disney stores is by far the majority of the crime on Disney property, which is similar to most major cities.
My first college program i was working in Merchandise in Adventureland. If you know the store that is connected to the causeway between Frontier/Adventureland(it was island something when i was working there in 96) then you can picture where this happened. Well i was training a new hire on closing duties and we'd already closed most of the exterior doors and brought in all the exterior merch. Two couples came to the only ajar door and asked me if they could purchase something. I said sure and had the new cast member put the drawer back in the register. Well one of the couples came inside and the other one, who were bikers(no judgements here) didn't. We took care of the husband that came inside and all of a sudden the wife said "hey, they just walked off with that planter!" The other couple decided they'd just take one of those Mickey wall planters off the wall and probably into a bag they already had. So i called security and gave them a description of the couple and they nabbed them near the train station on Main St as they were about to leave.
Also, a trick that people like to try is to buy film or disposables in one area, let's say over near pirates, and then go to your store and say they want a refund.
Thankfully there haven't been any huge tragedies on property as of yet. But cast members and guests are only human and unfortunately something like that may happen. But besides Disney Security and the handful of on property county sheriffs, there are also HRT's that are ready at a moments notice and all area/operations managers are trained for such crisis.
Bottom line, you should feel safe all over Disney property, just be aware of your surroundings.