I play my clarinet every day and when I’ve finished I remove the mouthpiece and hold it under cool running water for a few seconds before drying it with a soft cloth as thoroughly as possible. I don’t pull the cloth through but dry from each end of the mouth piece in turn.
Is this a good thing to do or am I likely to ruin the cork or damage the mouthpiece?
I’m sure that’s fine. If you have the option to let it air dry that may be the best. Personally, I do have a mouthpiece swab that I use. Sure, there’s an argument that it could change the shape of the chamber over time. But I bet the table changes regardless, and by the time the swab has any affect I’ll be ready for a new mouthpiece. Although I’ve had my clarinet mouthpiece for almost 20 years now (yikes!).
I have some vintage mouthpieces for my saxophones that probably date from the 1920s or 1930s and 1940s. They play just fine and I have no idea how well they were taken care of before I owned them.
IMO if you use some type of soft, smooth cloth you shouldn’t have anything to worry about. I currently been using the little clothes they sell for eye glasses. But I wouldn’t recommend using those mouthpiece brushes they sell with some of those care cleaning kits. Unless it’s really bad.
Running cool water through the mouthpiece shouldn’t hurt it. As for the cork it’s going to go bad eventually and they’re not that expensive to replace. Some water systems have chlorine in them. I don’t know if it will shorten the life of the corks or not. Either way a little cork grease should help prolong the life of the cork. I just put a smidgen of cork grease on the cork every time I practice.