The key is in the cleaning. No matter how good a player you are, the mouthpiece gets microscopic scratches that in turn harbor bacteria. This can eventually develop into a not-so-nice smell.
The best thing to do is literally scrub your mouthpiece in and out. You can get specially designed supplies at a music store, or just use a toothbrush. Use soap and water and you can even rinse with Listerine.
Run a cloth through the mouthpiece after every use and give it time to air dry if possible. Be sure your reeds are fresh (no mold i.e. Black spots) and air dry them too.
I would use a suitable brush which will reach into every little crevice and gently scrub it with peroxide of hydrogen. This is a pretty amazing liquid which is sold over the counter in pharmacies. It is H2 02, which is one atom of oxygen more than water, which is H2 0.It kills germs by oxidization.
You can use it to safely sterilize many things, including your toothbrush, counter tops, infected wounds and as a mouthwash. It will even help to whiten your teeth. Not knowing very much about saxophones, I would imagine that the mouthpiece is removable, in which case a five minute soak in peroxide will kill all bugs and leave it fresh.
If you use it as a mouthwash it will foam up while it is working, this is normal. The link below will tell you more about peroxide of hydrogen.
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