Inside the door are a couple of weld nuts - the nuts are secured to the inside of the door!
At the rear edge of the mirror's base, look for a piece of the plastic that can be pried up. Behind that secret door is a small set screw - probably allen headed. Loosen that screw and the mirror's base will release from a bracket bolted to the outside of the door. Tighten the bolts holding that bracket to the door then re-install the mirror and tighten the set screw.
I'm not quite clear on what part of the base you pry up and rather than risking breaking the base altogether I've attached some photos of the mirror from all angles so hopefully someone can point to what part to pry up.
The part of the base gasket facing the rear of the car has a small tip up point in the middle. Pry out this "tip" and you will see the metric allen set screw. It is about 1/8 in size, but is metric, NOT US standard. You want to locate a proper metric allen to use. If you strip out the head, it will be a B**** to try to get it loose or tighten it. Mine were sripped out when I got the car, and I had to eventually buy a used set to get undamaged set screws to tighten. As Eric said, the nuts on the inside of the door are welded on - stay away from them. Not only will they not solve your problem, but if you could somehow break the welds trying to remove them, you would not be able to attach the mirror to the door at all. They would have to be re-welded, with accompaning repaint of the door to repair the weld damage done to the paint.
After loosening the set-screw, place a towel on the door so you can hang the mirror on the door w/o scratching the paint, then inspect the part of the bracket that the set-screw tightens against - look for scoured material - this could be why it doesn't get tight enough.
You might have to remover the bracket and bend the tab that the screw tightens against down a bit so that upon reassembly, you enough 'tab' to screw against.