There are tons of bacteria in our tanks. Some are good (ie
denitrifying bacteria) and some are bad - like Vibrio or Mycobaterium. Not all tanks will have those bad bacteria, but you can pretty much count on there being something bad sooner or later.
Vibrio is a strain of bacterium that's similar (related) the the bacterium that causes Tuberculosis. You'd know if you had that going on - your fish would drop like flies (been there, done that).
Mycobacteriosis is a SERIOUS infection, that usually enters the skin through an open wound (ie playing in the tank with a cut). If you get this infection you can expect 6 months of irrigating the wound and taking some major antibiotics - it's no joke and it's very serious. It's relatively rare but knowing it's potentially there is a good piece of information.
Some folks are just sensitive and get redness of the skin and/or chapping from exposure. When Sally worked for me, her arms would get red every time she was in the tanks - said the same thing happens at home too (and probably all the time since she works at the GA Aquarium now!). That's no biggie really.
Pat your skin dry. Don't rub. Roughing up the skin opens the door to infection. Keep dry - don't drip dry, that causes chapping, and dry cracked skin. Use clean towels - if the towel was old and crusty and stinky, it's as likely to infect you with something as the water was.
SALT CREEP - WIPE IT! Many people contract these infections because they don't wipe the salt creep around the top of the tank before they start working in it. Top of their arm on the inside (which is soft and not toughened like your hands are) rub against the salt creep, which irritates the skin and can rough it up just enough for an infection to get in. The abrasiveness of the crusty salt is something that's easily avoidable if you wipe it down before you begin.
Having said all that, I have my hands in the water every day. I don't wear gloves... and perhaps I should (do as I say, not as I do!)... but I've not had any serious issues - but I do use caution and I am aware of the hazards so I'm as careful as I can be without using gloves. IMO having gloves that are wet on the inside can be just as hazardous - because a dark damp environment is an ideal breeding ground for bacteria...
Common sense - always common sense - and if you DO react to your tank - use proper protective gear.
Jenn