Cotton swabs and isopropyl (rubbing) alcohol should do the trick. Try to keep the alcohol from the plastics; use it sparingly.
There is also fancy contact cleaner used in the electronics industry that works quite nice and is very stable. For example, "CRC QD Contact Cleaner". Make sure you don't get the type used for lubricating potentiometers. We want the quick drying, stable variety that doesn't leave a residue. This stuff is made for cleaning sensitive electronics and metal connectors. When in doubt, read the datasheet for the product. It'll say things like:
Ideal for telephones, PCs, relays, edge connectors, tape heads, buss
bars, circuits, contacts, printed circuit boards, switches and circuit
This solution is also good for cleaning out really crufty carts, where the circuit board may be dirty enough to cause an electrical problem. In this case, you can either disassemble the cartridge and clean it gently with rubbing alcohol, or (if the cart is hard to split without breaking it) you can use the contact cleaner to spray inside the cart, letting the liquid run and "boil" (it has a very low boiling point, which is why it works as it does) clear out the cracks in the cart.
Bonus: Good for cleaning keyboards, too. Especially older chiclet style keyboards.
It should be ok for paper labels, etc. to be wetted with contact cleaner, but try not to soak it.
However, unless you really have to, you probably don't need to clean out the inside of the carts. Maybe if they have been moldering in basement for years and you want to get rid of unsightly splotches or visible dirt. Electrically, everything inside the cart has a good connection (or did at the factory) and cleaning will not help improve that. But, as I say earlier, if cruft is causing what might be an electrical short, causing dodgy behaviour, cleaning the circuit board carefully with the right stuff is fine.
The outsides of the cartridges should be treated the same way they told you in the original packaging material: at most use a damp cloth with regular water (and maybe a little vinegar if you want to freshen them up) and allow to dry.
Don't use a rubber eraser. This eventually removes the gold or copper cladding on the contact fingers. In a shop I worked at, you'd be read the riot act if you rubbed contacts with anything. And I'll have to disagree with anyone who suggests that paste metal cleaners from the store are useful for anything but the grungiest, filthiest carts, and even then I wouldn't recommend it. These compounds often have gritty lapping material in them, which is all sorts of bad.
I've also seen people recommend sandpaper for stubborn oxide. This would be a mistake, and should not even be considered a last resort. Under few circumstances would oxide be so crufty that it needs grit to remove (and grit causes scratches, which encourages and promotes further harder to handle oxide). If your carts are this fouled, you are already into Retrounaut Archeology, in which case you are sort of on your own. The rest of us should stick with non-destructive or minimally destructive techniques.
And, yes: moist air from your mouth is not great for metals. I understand that blowing into the cart is part of our learned wisdom for getting these things to work, but it's really selection bias at work.