I would suggest teaching your wife how to play gin rummy or something - much cheaper plus how are you going to know where and when it is going to rain so you can have your trailer there?
The cars are 13 feet long - so I guess a 14 footer would do it but there are lots of other questions that come up - usually when people have an enclosed trailer they also like to take OTHER stuff - like lawn chairs and jacks and tool boxes and....well you get the idea. I would recommend a 20 footer - not to big and not too small. Plus easier to sell when the time comes.
As to the advantages of an open trailer vs. an enclosed trailer....well with an open trailer your car is still going to get wet so you will find yourself parked under that same bridge with your wife...see my earlier comment about gin rummy. With an enclosed trailer you can sit IN IT under the bridge and play gin rummy.
An open trailer is usually lighter so you don't need a big truck to pull it....but then again you can't easily put the lawn chairs and tool boxes and stuff on it either....
Pace makes a good trailer - had one of those. Buy it used.
Also....make sure you put some good tires on it! The Koreans seem to have the trailer tires market all sewn up - Nankings or something - they work way better than the Goodyears.
Good luck!