Good advice on the enclosed trailer.
You asked specifically about a trailer, but I used to drive a 1 ton Ford van with a box on the ass end. The entire "shop on wheels" was totally enclosed with the welders' exhaust piped out through the bottom. This rig had one of the older generation Trailblazers with the HF for TIG and about 200' of lead. Despite the physical size of the machine, it was fitted so oil changes, maintenance, etc. could still be performed inside the box. The gas line was hooked into the trucks two gas tanks.
We could SMAW, FCAW, GTAW and GMAW with the same machine. Used it for cutting when I was too lazy to drag out the gas hoses. We would go out and weld in the rain (inside the box of course) when other welders were sitting waiting for the weather to clear.
That was 1984-87. This guy still has the same rig, excellent shape, same Miller Trailblazer. Custom paint, it still draws attention.
The biggest advantages to this set-up is:
1.) You only lock the vehicle doors to prevent theft. Not a bunch of tool boxes.
2.) You can work inside the box on smaller parts during inclement weather.
3.) ALL of your equipment, consumables and materials could be stored in a secured, dry environment.
On the other hand, I have seen heavy duty rigs (Mack, HD Fords and Chevys, etc.) with rigs on the back that look very sharp. But, the equipment is still exposed to the elements. I think the Trailblazers have some protection like SS covers and what-not. But it does not keep out any fuilds, rocks, bugs, birds (no kidding), etc.
Trailers increase OA vehicle lengths, restrict access, difficult to back up depending on what is pulling it, etc. Several of my clients have had their trailer rigs ripped off. One had 3 rigs stolen in one year. Unless your good buddies with the crane operator and he'll hoist your rig over the weekend, I would not leave anything for the thieves.
Sorry for rambling.