I repaired my antenna last week on the 81 I just got. In my case the antenna would also not go down, however, it would go up about 3 inches if I pulled on it. I decided to pull the thing out and see what was wrong.
First off the retaining nut on the top was a bit hard to get loose. I don't know if there is a proper tool to use on this nut but I used a set of needle nose pliers with the tip of the jaws bent 90 degrees. The jaw tips fit well into the slots of the nut.
The bracket at the bottom can be removed from the antenna to get a little bit more room to get the thing past the muffler.
Next I needed to disconnect the wires from the antenna motor. I found that there is a connector behind the drivers rear speaker. There is also a connector for the antenna wire in there. After pushing out the large grommet from the inside the wires,grommet, plug and antenna could be removed from the car.
I found that the cable had broke, this is why the antenna could go up but not down. The motor had also been disconnected at that connector behind the speaker that is why the motor would not turn. I can only assume that the previous owner had disconnected the motor to stop it from continuously running as that is what it would do with the broken cable.
I found 3 wires coming from the motor-- white,gray,dark green. If you put 12 Volts on white and ground on green the motor should turn in the down direction. You may be able to stuff the antenna down with the motor turning the proper direction, but it might be better just to take the whole thing out since you have already got this far. It will give you a small project for the winter as parts are available from corvette parts sellers or maybe even GM.
As for mine I was able to repair the cable end for nothing.