Not that much to worry about.
First, can people get to the storage units without some type of security system? Some places have a gate which can only be opened with an access code. Others don't, and anyone can drive right up to the units. Obviously a gated/access-controlled system is better security for you. You are responsible for insurance. Some places are actually inside buildings - even better, since it would have better climate control. You probably need to give some consideration to climate control unless you're only storing a rock collection. Outside units that will endure high heat and humidity can mess your stuff up.
Compare long-term rates to the month-to-month rates.
Use pallets or something to keep your belongings off the floor in case it gets wet.
Think about what you might want to retrieve out of storage and keep that near the door so you don't have to unload the entire unit to get something.