I have an '89 but honestly have never repaired one on a 'vette. I have done them on other cars (including my '85 Riviera and they all follow the info provided in the following website
www.miata.net/garage/power_antenna_repair.html
The key is to make sure that there is nothing broken off in the antenna motor or channel and that the links are drawn all the way into the channel. If the motor is tossing out the antenna it is not shutting off correctly. On the older cars there was an adjustment screw or there was something on the link that tripped it off.
You may have to get an antenna replacement--I understand that Zip sells parts.
Good luck, I hope this helps.