February 2016: I found that the backup lamps weren't working when I brought the car home. After consulting the factory service manual, I located the backup lamp switch at the rear of the transmission. When I tried to disconnect the wiring harness from the backup lamp switch I found the WeatherPack connector was fractured and it appeared to have been repaired with Super Glue. Clearly the connector would need to be replaced.

On to the switch. When I shorted the two wires in the connector the backup lamps turned on. So the switch was defective. I ordered a new switch from Ecklers:

Corvette Back-Up Light Switch, For Cars Manual Transmission, 1984-1988 25-112412-1 $31.67

Direct link to this image:
https://corvette.westhaver.com/1988/backup_lamp_switch/image_001.jpg