Hygiene is always something that woodwind and brass players should be aware of and I actively encourage my clarinet and saxophone students to wash their mouthpieces at least once each week.
All reeds and mouthpieces can be easily washed in cool water (and dried immediately) and most household wipes and disinfectants can also be used. Make sure you always read the label and obviously don’t use products that are not safe to be put in the mouth (e.g. bleach!). Antibacterial mouthwashes are also very effective at cleaning reeds and mouthpieces (and making them taste nice too!).
Metal and plastic mouthpieces can also be washed in warm soapy water (e.g. washing up water) with no problem, but please be aware that this might turn ebonite mouthpieces a little green! As a general rule, if your mouthpiece came with your sax then it's almost certainly plastic, however if you bought your mouthpiece separately and it was reasonably expensive (around £100 or more) then it might look like plastic but is in fact probably made of ebonite (a type of hard rubber) and might turn green in warm, soapy water.
One of the best and safest ways to clean any mouthpiece is by submerging it for half-an-hour (or longer) in a solution of half white vinegar and half cool water. Simply fill a glass with the mixture and drop in the mouthpiece! Scrubbing a mouthpiece with a cloth soaked in wood soap oil is also a good option for removing stubborn dirt and grime.
Flute head joints can be washed in similar ways, however they have a cork inside that shouldn't be allowed to get wet. Wiping the lip plate and embouchure hole with a cloth soaked in anything mentioned above should be fine, but try to avoid letting the inside of the head joint become wet.
Lastly, never submerge the body of your instrument in liquid as this will ruin the pads and cause the whole instrument to stop working. If necessary, a technician can strip off the keys in order to soak the body, but this shouldn't be done with keys still attached.
Please note, the information provided on this page is based on my own extensive knowledge and research but please consult a qualified instrument technician or healthcare professional for further advice. Anything you do to your own instrument is your own responsibility!