Obviously security and access, but #1 is surveillance. In a busy storage facility it's more common for batteries and LP tanks to go missing and/or some new guy backing his trailer for the first time backs into yours. If you plan to go out frequently ask about access to electric so that you can plug the TT in and leave the fridge on. If not consider a small solar battery maintainer (which may grow legs...)
Some lots only have a camera at the gate so if your TT gets stolen they can see the truck that towed it away, but anything else is "off the record". I would want 24/7 cam coverage so if anything else goes awry there's a record of it.
Also consider joining your local camping/campers FB page or community page and ask for recommendations from other RV/boat owners. Any of your neighbors store an RV or boat somewhere?