NES Console
72 Pin with four different choices: Boil, Clean Bend, or Replace
Clean
You can clean the connector somewhat by putting isopropyl alcohol on a clean cartridge and reinserting it a few times. Clean the cartidge between insertions. Or, dismantle the console and clean the connector itself off of the board.
Boiling
While this sounds like it couldn’t possibly work, it looks like it does.
Bend
I’d really recommend against this. I’ve tried it. It’s hard to do and I did more harm than good. If you must, get some quality dental tools. If cleaning or boiling didn’t work you should replace it.
Replace
You can get a replacement for between 10 and 30 dollars on the internet. Here’s what 10 gets you.
I’ve tried one of these before and didn’t like it. Super stiff and felt like it was scratching the pins of the cartridge.
The Blinking Light Win, on the other hand, looks very promising.
Go more expensive if you can. The cheaper ones are often not built to nice tolerances. Replacing it is a matter of unscrewing screws and careful popping off the old one. If you’re reasonably handy it’s extremely doable.
SNES Consoles
The easiest way to clean the connector inside is to clean a cartridge, then wet the cart, insert it, clean the cart, and repeat that process. An alternatively is to, very carefully, use a microfiber cloth and alcohol. There’s a lot that can go wrong (snag on the pin because you delved too deep) so be careful!
The design of the SNES is such that the pins will rarely go bad, which is good because replacing the pins is more or less impossible without a lot of soldering.
Note on replacing batteries
Games with save systems rely on a battery backup to keep the save stored on the cartridge. That battery will eventually go bad. The batteries are a standard CR2032 and is–relatively–easily changed. You’ll need access to the insides of the cartridge. After that, you disengage the battery from the cartridge , replace it, and secure the battery back in place. This might require some soldering, although there are tutorials for ways to do it without soldering. You can also buy battery replacement kits