I store mine in the garage with the cooler running :B
However my trailer has to stay outside. Like has been said already the plastic is the one of the first to go along with any unpainted wood. Glues, calking, and seals will dry out and harden. Painted surfaces will fade and gel coat will oxidize.
In my opinion, the tires are really an issue. For the last 9 years keeping them covered, I average a set of tires ever 2 years on the trailer. I will have one come apart so I replace all of them. On the last set I removed them and put them in the garage. We'll see if that helps.
It's just better all around to store inside in the summer.