I've been going to WDW with just me and DD since she was 5 and I can honestly say that I've always felt safe and had a great time. WDW is one of the easiest one-parent trips going.
You can give the hotel most of your cash when you check in, then charge things back to the room and have your deposit cover it or you can lock it in the safe and take it out as needed. I always carry a small amount of money, one low-limit credit card, and the room key to charge with. It works out well and there's nothing too stressful to lose.
As far as safety, I tend to be a worry-wart on that score, but I can honestly say I've always felt safe at WDW. I mean, you don't lay your cash out on the bench next to you or have your credit card poking out of your back pocket, but keep your eyes open and you'll be fine.
Have a wonderful trip!