My remote has quit working however the red light still shows when a button is pressed. Can I reprogram it?

The red light means the transmiter just transmited, the receiver is apparently not receiving. Unless you changed the dip switches on either unit you will not be able to reprogram this. My quess is that your receiver is shot. This link gives Hampton contact info. https://www.askmehelpdesk.com/electrical-lighting/ceiling-fan-troubleshooting-repair-frequently-asked-questions-46738.html

Check the dip switches first, sometimes they get displodged. Make sure they match, and try a different setting.

Ceiling fan remote controls - help and FAQ- Ceiling Fans N More (http://www.ceiling-fans-n-more.com/ceiling-fan-remote-controls.php)