For most, a little red ink here or there on a piece of money that you will only give to someone else in exchange for something you really want doesn't really mean much. To the avid collector, however, red or blue ink on a rare note can significantly reduce its value. For these collectors, the challenge is finding a paper money cleaning method which is strong enough to get out ink yet gentle enough to not damage the paper.
Practice the amount of pressure to use with a toothpick on a note of lesser (or no numismatic) value first. This can be an old one dollar bill.
Launder (literally) the money with regular clothing. Tie it in a cloth bag first. This is not advised for collectors, but it has been used by the Treasury Department as a way to clean soiled bills. Afterward, the bills were pressed, dried and returned to circulation.
Use hairspray or apply rubbing alcohol on the opposite side of the ink mark. Press the bill down on a wet white cloth. Slowly outline the ink stain with a toothpick; do not press hard. Use the toothpick as a way to apply specific pressure; do not rub.
Use Oxyclean detergent using the same method in step 3. This works well and is easy on paper. Use a paste, and apply with a toothpick.