What should I do about this possible mold I found under paint?
IMO you are making several leaps here that don't fit.  You've got mold below a window, spread out about 2 feet on either side, and you suspect condensation as the culprit.

You specify that the damage is old - years old by the look of it.  Condensation happens every year, every season.  Why would the damage be old, it should be fresh if it's condensation.

Finally - if condensation were the culprit, the condensation and thus the mold would be uniform. Water leakage through a window, however, is going to fan out below the window in both directions...

The answer is your window leaked at some point - years ago by the sound of it since the damage is old.  

Solution?  

Remove and replace ONLY the section of drywall that is damaged.  Do not add a partial vapor barrier as that's about as useful as putting in an aquarium with only 2 sides.  Mud and tape properly and repaint.