Do NOT blow on your old carts. The moisture from your saliva CAUSES the oxidization of the cartridge contacts. I generally use a q-tip and some low percentage isopropyl rubbing alcohol to clean contacts. Just dip the q-tip in the alcohol and dab off the excess on a paper towel and rub it along the contacts in the cart. Then used some canned air duster and the other side of the q-tip to dry it off.
If you want to clean the socket in the SNES, you can either buy one of those foam slot cleaners that they used to sell for that express purpose, or what I use is a butter knife and a clean tshirt. Dip a little isopropyl on the tshirt corner, you just want it damp so you don't drip into the socket. and then wrap the shirt edge around the knife and push it into the slot. don't force it too much because you don't want to bend the pins, but it should slide in and out fairly easy. I got a lot of grime out of my SNES slot the first time I did this. Then blow some air duster in there to make sure it's all dried out before you plug it back in and turn it on.
The classic answer to this is to blow in the cart where it goes in the console. This is good practice for cart-based systems, I suppose, it does the trick with the N64. If this doesn't work, the contacts on the cart might be particularly oxidized/dirty, try going at it with a toothbrush and some contact cleaner. Avoid getting too much contact cleaner on anything plastic, as most contain acetone and will warp plastics.
E: Huh, didn't know that, Andrew. I though the moisture increased the conductivity of the pins. Never thought of it in terms of oxidation...may
sometimes the awnser is more difficult. Repairing the Super Nintendo i encountered several problems of which bad capacitors and a corroded reset switch are most common.
The caps used in the 90's aren't the best ever. Since you can't look underneath them you need to 'wiggle' the cap to see if it is loose. If so the legs could be corroded. but, there are only a fev capacitors on the board so i say repace em all for a buck or 3 to be sure.
Second is the reset switch. Normally when you press it you short out two contacts. When the switch is corroded or broken, the swicht makes contact all the time and therefor it cant start because it is resetting all the time.
There are several other options but these are fairly easy to check.