You can go to Radioshack and get some electrical cleaner. Rubbing alcohol works alright, but you won't want to pour it on there. If I was using the alcohol, I would take the switch apart and see if I could get into the smaller parts. Then you can use a dab of white grease or the grease that they use in small plastic parts. Just a thin layer should do. If you take your bike to RadioShack they should be able to help, but they are no longer an electronic parts store so it's a shot in the dark really.
A really common problem. If you search this forum for "sticky switches" you will find all sorts of remedies. All easy to do. It's not just the mute switch that suffers with this problem it can happen to them all so it's usually best to separate the switch block and give them all a clean with contact spray followed by a spray of silicon grease. When you search you will also find pictures other members have posted which helps if you have not separated the switch block before.
The GL1800 seems to suffer more from sticking switches, and sometimes they need to be opened up. Which Goldwing do you have? It makes it easier for us to help if we know which Goldwing you need help with.