How to store fresh figs

There isn't that much more you can do. Fresh figs are perishable items, and before you get them you don't know how they've been stored. Moisture is your enemy here, the drier they stay the longer mold will stay at bay. The containers you buy them in often trap moisture so I would remove them from that packaging and store them in the refrigerator, on a shelf, uncovered.

Also, keep in mind that a bit of mold outside does not mean spoiled inside, I know it is unattractive but you can still cut them in half and scoop out the good stuff, provided that is still fresh enough.