They should be protected already as they lived 25 years in this ugly world, so not extreme measurements should be needed. Shorting the output contacts might drain the battery as you don't know how they are connected, but it will keep them safe, so for games without a battery might be fine.
For games with batteries, yoou could look further to what you need or keep them connected to something wich provides ESD protection, but you would need a lot of connecrors to keep all your games protected. One option could be to have them in metalized bags as the ones components come in, and good be good to provide some grounding inside your plastic box, so all stay at gound potential and when you take them out aren't charged.
You should be careful while handling, that's the most risky moment, keep your consoles grounded, and ground yourself while handling. So now your ganes are stored at ground potential, your console is at ground potential when connecting the games and also are you while handling. Bare foot and antistatic floor mat or conductive floor could do. If you don't want to be bare foot, there are ESD safe shoes, ESD safe shoe accesories, with a band from inside to the floor or the classic wrist strap wich is kind of ok working on the bench but not so much to move around.
JS