If you're satisfied with the security features of the storage facility, I wouldn't think your trailer would be at any greater risk for break in than storing it at your home. I have been using an RV and boat only storage facility for 5 years, located 12 miles from my home and have had absolutely no problem. The owners have every reasonable security measure they can employ on the property and I maintain insurance coverage on my trailer. About as much as anyone can do.
I will say my insurance premium is slightly higher because the trailer is not stored at my home. Yes, even though locks on an RV are flimsy at best and can be easily defeated with not much more than a screwdriver, you should lock everything and pull your battery, not so much for security but if you don't have power available in storage to run a charger on the battery, you could do so at home.