You don't give us enough information for a detailed answer, but here's some very basic battery hookup advice.
Be sure you know which is the positive cable and which is the negative, and don't try to hook them up backward. Positive and negative poles on the battery should be clearly marked. You should be able to find where the negative cable is connected directly to the trailer's metal frame, if you can't identify it otherwise.
Always, always hook up the positive side first. That way if your tool accidentally touches the metal frame of the vehicle, nothing happens. The same logic says to disconnect the negative terminal first when you're taking the battery out.
If you're going to work on the DC wiring, disconnect the negative battery terminal and remove power from the univolt. 12-volt DC isn't enough to hurt you, but it can start a fire if you accidentally short something out, not to mention nasty surprises.
As to your blown fuse, something has caused too much current to flow through that particular circuit. Might be a wiring error, short due to insulation rubbed off against metal while one the road, wires chewed by critters, who knows? If I were you, I'd leave the fuse blown until I had been able to install a battery. That should isolate the bad circuit until you can find the trouble.
If you don't own a volt/ohm meter, get one. Find an affordable balance between high-priced professional quality and cheap junk. Some of us here can talk you through using it to troubleshoot your trailer's wiring.
And, best of luck with your new Airstream!