OK, let me start by saying that maybe I am not the very best about cleaning my instrument. I do it consistently, just not every single time I play.
Maintenance is super important to the longevity of anything—your instrument, your gear, your body, etc. Take care of it, and it will take care of you.
Let’s start at the top. The most important part of the saxophone is the mouthpiece.
If you have a hard rubber mouthpiece, NEVER use hot water. Cool to slightly warm is fine. If you use water that’s too hot, you can warp, discolor, and irreparably damage the mouthpiece.
Use lightly salted warm water in a bowl, a small toothbrush with soft bristles, and just a pinch of toothpaste. Keep the mouthpiece under the water while you gently clean the inside and the outside of the mouthpiece. Keep a good grip on the mouthpiece, so it doesn’t fall and chip.
Using a little toothpaste will help keep your mouthpiece smelling fresh without putting any harmful detergent or chemicals into your mouth. If you didn’t brush your teeth for a week, think how that might smell. Your mouthpiece has the same stuff in it.
You can also run a silk swab through the mouthpiece after each playing.
The next most important part to clean is the neck. Germs breed inside! Get a neck brush from a music store. Every few days, run some warmish water through the neck and use the neck brush like you would clean a bottle. You might be shocked the first time you clean this way. Swab inside when you’re done and dry the cork.
You should be careful to play with a clean mouth; no sugar or food right before you play as it gets onto the pads and makes them stick more than usual.
No need to put a huge, furry swab into the body of the horn. It can do more harm than good.
Good hygiene will benefit you and your horn immensely. Happy sounds and happy smells!