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.
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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...maybe I can get some alcohol in a spray bottle and some paper towels for my cart systems instead of blowing in them. Thanks for the tip, I guess.
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.