My answer would depend on two factors: how often do you use your trailer and what is it made of? A cover would protect a trailer from the elements, but if you are using it weekly, could be more trouble than it’s worth. If you use your trailer seasonally, consider storing it inside --many here do with RVs, boats, and horse trailers --the fair grounds charges $100 for six months under-roof storage. You would not have access to your trailer at all.
If your trailer is steel, I am not sure covering it would make much difference slowing the rust issue. If it is aluminum, then you would have less cleaning to do. But be aware in my opinion, the BEST solution to keeping an aluminum trailer looking new is frequent washing and waxing. My 2004 Merhow (aluminum) trailer sat outside for 14 years, on grass. I used it weekly. I washed it weekly (ok, not in the winter, but if the hose wasn’t frozen, I washed it). When I sold it two weeks ago, it still looked new --no rust/corrosion, no black streaks, no stains --oh, I bought another Merhow.
My white trailer sat out under trees --no protection at all.
But, --not everyone has a husband who is a classic car buff and knows every product to use to keep a vehicle looking great, or a place where they park by a hose every time they use the trailer.
Still, my answer is washing/waxing is better than covering --but inside storage is best if you don’t use the trailer.
And since you are reading this —have your tires checked every year, replaced every 3-5. Always have a fire extinguisher in your trailer (and truck) --$12 could save your horse’s life.