I regularly tow an enclosed car trailer behind my diesel pusher. Other posters have already given you some great ideas about what to consider.
-- In my opinion, getting the extended tongue on the trailer is one of the most important considerations. It helps make sure you don't pinch the front corners of the trailer against the back of the motor home during tight maneuvering situations.
-- I use tire bonnets to strap down the car inside the trailer. Very secure and unlikely to damage undercarriage components on the car. If you're going to tow the same car all the time, then properly placed D-rings are probably the least expensive option. I use the EZ track system which allows you to vary the attachment points depending on the specific load.
-- I had a 12v deep cycle battery installed in the trailer. Also dome-type ceiling lights inside the trailer and an outside spot light over the rear cargo door and ramp area. Really helps out during night-time loading and unloading situations. Also, had a cigarette lighter type power port inside the trailer.
-- Getting in and out of the car when its inside the trailer can be a hassle. I have a small convertible roadster so its no sweat with the top down. Depending on what you plan to put in the trailer, you may want to test fit your car in various models of trailers to see which is the best fit. Inside my trailer for example, there are box-like enclosures around the wheel wells of the trailer. Once my car is parked inside the trailer you cannot open the car doors at all. For me, its not a problem. But,if you planned to exit the car by squeezing through a partially opened door, you couldn't do it.
Hope this info is useful. Good luck!