Trappers skinning critters have the opportunity to catch several diseases. Tularemia, echinococcus, Streptococcus, and rabies are all possible diseases you could catch.
A friend did research on the prevalence of trappers getting exposed to rabies while trapping arctic foxes. He found several individuals that had never had the precautionary shots and still had a antibody in their blood. They got from skinning the foxes. They obviously did not die. So yes, you can get rabies from skinning a critter. The incidence of people dieing is low from skinning but it happens. I agree with traprjohn on the precautionary shots, if you live in rabies country.
Several people I know have developed really bad strep infection sfrom skinning, in particular wolves and brown bears. Both had significant doctor bills to get over it and one guy lost use of the infected finger.
Enchinococcus is something you can contract from messing around with scat and stomach contents. It's found mostly in canids from foxes up through wolves. It's bad stuff.
Humans can get tularemia by handling animal skins, and eating meat from animals infected with the disease.
There is some great information on the CDC website at
http://www.cdc.gov.It' safest to just wear rubber gloves for all your skinning. I know some people who have gone to wewaring rubber gloves and then thin cotton gloves on top to help improve the grip and for a little more protection so the rubber glove is less likely to break.