Yes, I
always wear gloves when cleaning. I wear them to prevent my skin from absorbing chemicals in the cleaners I use. I have dry and cracking cuticles, especially in winter weather, so I need to keep solvents away from my skin as much as possible. I even wear them when reloading, but for a different reason. During reloading, I wear them to prevent skin oils (the little I have left) from getting on cases, primers, and powder.
A link to the gloves I bought is
here. These cost about 42 cents/pair and I can get several cleaning sessions out of one pair before I toss them. I recommend heavy duty (which means thicker, like 8 mil) nitrile only. I also recommend getting a size that fits somewhat snugly on the had to allow better tactile feel when handling small parts. I do not like wearing loose fitting, wrinkly gloves.