Wear nitrile gloves and have hand sanitizer on hand. This is what I do, and who knows what diseases all the critters I have skinned may have had all these years doing taxidermy and trapping. I have never gotten sick from any of them.
Rabies is usually transmitted through a bite, so don't let a live trapped animal bite you and break the skin.
There have been rabid animals in my area too. When I walk around the property I am armed. Of course since I started doing this, there have been no rabid animals, no pitiful mangey foxes limping around, no foxes in the front yard banging against the fence after a night of guinea killing, and no catching any foxes in the actual act of poultry depredation. Some sort of .40 cal magic talisman or something...LOL
Many years ago I was walking on one neighbors property, when I encountered another neighbor. The attraction was the beaver pond down in the woods. He was with a young kid that was riding on his shoulders. He also had what looked like a .22 rifle. I asked if it was for groundhogs, just joking around, but he wouldn't answer. I was worried he was shooting the beavers, but he tells me how he loves the beavers and has videotaped them many times. It took me several years to figure that he was probably carrying that rifle due to the rabid animals. Open carry of long guns is legal in MD w/out "gov't permission slip".