Ideally, in acid free storage boxes with dessicant packs and a dehumidier running with hygrometer to monitor the relative humidity (this is how things are stored in museum archives). All this stuff is available from conservation/archival/museum supply places online, like Carr-MacLean in Canada.
Of course, this is not realistic for most people storing stuff at home. In my case, I just bought some nice large rubbermaid bins with snap-on lids. They do the job well. They can also be purchased in smaller sizes.
If this is beyond your budget, then perhaps the individual game boxes can be stored in large Ziplock bags (you can get very large ones). You might also want to take a sharp point such as a screw or nail and poke a few dozen pinholes in the Ziplock bag so that there is some atmospheric exchange, but small enough that drips of water will not penetrate.
If mould is really a problem, even with the rubbermaid bins, then the dehumidifier might still be a good idea....but it sounds like in your case the mould was started because water dripped directly on the items in question. The rubbermaid bins should prevent this.
