If you use Magic Eraser of fine grit sandpaper, just be sure to tape off the blank ahead of the cork, and the reel seat behind the cork, so when you slip you will not damage either of those with the abrasives in them.
Why are you cleaning your handle? Is there some sort of stain on it?
I've only ever wiped off my cork handles with a damp cloth, being careful to make sure they're completely dry before putting them back in the sock and tube. I like some "patina" on my handles--don't want to clean them up too much.
Unless things get slimy, I do my rods once a year with a squirt of dish soap in a bowl of water and a sponge. Quick rubdown on the handle. Nothin' fancy.
I clean the cork on my salt water rods as they get covered in fish slime and other ****.Hot soapy water usually does the trick.Sometimes use those foam Magic Erasers.
Once a year or so I wash my rods in the tub with warm water and a couple pumps or mild hand soap. I like the cork to be nice and worn in though so I don't spend much time cleaning them. The guides however, get a thorough cleaning. As mentioned above, be sure everything is totally dry before sealing them up in the tube again.
I like my cork handle clean. Easy enough to rub down with wet wipes every few outings and it will keep the cork in excellent condition. Make sure you air dry it before putting back in case.
Warm water, Dawn and a sponge (not the green abrasive side). Light pressure, the filler they use seems to scrub away pretty easily. If you're getting ready to sell the rod, give the blank a once over for water spots, finger prints, etc...