So my college's bassoon (a schreiber) has stain coming off of it. I just noticed today after completing my jury. In any case, should I be concerned? It's probably just humidity, but I have no idea. Also, does anyone have any tips for getting the stain off of my skin? My fingertips are now dyed red. I tried soap and water, as well as rubbing alcohol. I don't want to jump right into chemicals if I don't have to. Thanks all.

That sounds as if the varnish has worn away leaving the base coat of stain exposed. The varnish can be repaired where necessary by a competetnt technician. As it is a college bassoon thell them - repairs shouldn't be your responsibility.
Regarding the stains on your fingers it should disappear after a few days better that imo than using nasty chemicals. The trouble is if the varnish isn't renewed the same is likely to happen again during a heavy playing session.

Varnish often wears off around the open finger holes on the wing and boot joint, particularly left hand first finger E and right hand first finger B flat. These two tone holes are angled downwards so that water tends to find its way out there. The stain won't do you any harm, but the tone holes should really be re-varnished to protect the wood (and stop you getting red fingers!)