first the window AC must be set in the window at an angle to allow the water to drain on the outside of the unit on the furthest end of the outhanging unit. Set a small torp level on the top of the outide of the ac and be about one half bubble toward the house. This fall should allow the water to drain away from the sill.
If this is a single or not an ongoing leak, the studs should be OK. Opening up the wall and drying it out is your call. But if its not a long leak then don't completely worry.