Ask when you get there, but something tells me that you're probably safe regardless. Thieves ply their ugly trade anywhere, but Disney cruises are probably safer than just about anywhere. The people are probably a bit more law-abiding than just about anywhere outside of a convent, and they're well-heeled financially to be able to be there in the first place, thus less likely to steal for a living or hobby.
Here's what I would do if there are no "secure" areas:
1. Take a towel.
2. Get it dirty with sand and sunblock and other gunk.
3. Wrap your valuables in the towel.
4. Leave the bundle on a nearby lounge chair.
5. On top, put some "intimate" clothing or other items such as sanitary pads in a clear plastic bag.
Trust me, nobody will want to touch your stuff. Especially men, who are the more theft prone of the species.