Rubbing alcohol will NOT harm your games, period. It's a highly recommended and easily available cleaning solution for pretty much any electronic device, it evaporates quickly which reduces the chances of corrosion (if there's any chance at all really). I've heard mixing in a bit of water helps, I'm not even sure if that's true or not so don't take my word on it.
Take some rubbing alcohol and some Q-Tips (cotton swabs), dip the Q-Tips in the rubbing alcohol and rub the edge connectors on the bottom of each cart. Depending on the dirt concentration it may take several Q-Tips. Keep cleaning until no more dirt comes off the carts and make sure to let everything sit for about 5 minutes to let the alcohol evaporate.
Do you have this problem with your other carts and not just the ones you recently got off eBay? If so, take an old toothbrush and rubbing alcohol, dip the bristles in the alcohol and lightly scrub the cartridge connectors inside your AES, be delicate with this, try not to drip too much inside the connectors and make sure everything is dry before testing.
BTW, if that bundle is entirely cart only I think I saw that lot on eBay before, kind of funny I'd see who actually won that auction.
EDIT: I noticed you mentioned an eraser. Combining an eraser and rubbing alcohol/Q-Tips is really good for cleaning cartridges, I personally use this method.