What you have is called a bubble skylight. The older ones are notorious for being poorly insulated. The moisture builds up in there because there is a big difference in temperature between the air in the room, and the surface of the skylight. It is basically acting like a chimney, and causing you to lose a lot of heat throughout the year. A poorly sealed skylight is nearly as bad as leaving a window open a crack all year.
What you need to do to prevent this from happening is to insulate the skylight. The cheapest, and most effective way of doing this is by installing insulating window film. These kits can be found in most home stores, and are pretty inexpensive.
Inside of the box, there is a roll of window film, and a roll of double stick tape. It is pretty simple to install. Tape around the perimeter of the window first without exposing the other side of the tape. Measure the film, and cut it oversize by a few inches. Peel off the plastic from one piece of the tape and position the film over it. Then you can peel off the plastic from the rest of the pieces of tape and carefully install the film. You want to avoid any creases when putting it on, and make it as straight with the window as you can. Then take a hair dryer, and gently heat up the film until it begins to shrink, which tightens it up. You want to avoid heating it up too much because it can shrink to far and separate from the tape. The goal is to make it completely air tight. Once the film is tight, you probably wouldn't even be able to see it most times.
That should solve the problem with moisture building up. You may occasionally get a little bit of moisture, but it shouldn't be nearly as bad as it is now. You will probably also notice that there is less of a draft, and your heating bill should go down a little.