I have an open trailer and an enclosed (6x10) with a rear and side door. I paid $2300 for the enclosed trailer from Tractor Supply about three years ago. I happened to mention to the store manager that I was looking for an enclosed trailer and he had one new one he wanted to get rid of as he usually only sold open utility trailers so he reduced the price. NY registration on mine is about $22 a year with a $6 annual inspection. The car insurance covers the trailer so a policy for the trailer is not required.
Since you are not going to use it much I suggest you look on Craig's List for a used one. Might save you quite a bit over buying new.