If you are dealing with a fiber optic cable, all you need is a pair of work gloves just like ones you can pickup in any hardware store I can think of. If you are dealing with the fiber itself - stripping, cleaving, terminating, splicing - you're better off with your bare skin. Can feel much better and your fingers move easier -so less chance to break it. If you are careful, there is not much chance to get the glass splinter you're probably concerned about. Even in gloves, the glass is so thin that under a right angle it will pierce just about anything.
While the problem does not seem widespread, I have heard horror stories about bits of fiber piercing the skin and getting into a vein, then causing major health problems. That stuff is no joke. Same thing can happen with hypodermic needles, if broken off in the body they can cause SERIOUS injury and death from moving through the blood system.
Again, it doesn't seem to be a widespread problem with fiber but the fact that it can happen scares me. I always try my best to find a broken off piece of fiber, it really bugs me to not find it-later in the day at home I will rub my eye or something and then get paranoid about that piece of fiber somehow being stuck to me all day then getting in my eye. Yuck.
These small particles are very dangerous. People who work with carbon fibre material must be very carefull to prevent thin fibres entering the skin as this stuff is brittle and can enter the circulation system with bad results. J McC