Unfortunately figs are not long keepers.  The dryer and cooler you can store them the longer they will keep.  It is recommended that you not wash them before refrigerating and also try to keep them as dry as possible.  You might try placing them on some paper towels in plastic bag that you can seal to prevent moisture from getting on the figs.  They can mold easily.  The crisper works well.  

Figs should be picked ripe but not soft and mushy.  These will mold and spoil faster. They do not ripen well if picked green.  Also do not store over ripe fig.  They have a short storage life and can spoil the whole batch.  Keep away from apples and bananas that put off Ethelene gas that ripens fruits. 

If you have a lot of figs I suggest you preserve them by drying, freezing or canning.  This handout has suggestions for preserving them for future use.  
https://extension.oregonstate.edu/food/preservation/preserving-figs-sp-50-648

Hope that helps.  Thanks for using Ask Extension.

Nellie Oehler