How to treat mold under the caulk around the kitchen sink

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.

Greenboard is sheetrock where the paper is water resistant (but not necessarily water proof). A better alternative to greenboard is to use a paperless sheetrock product like Densarmor, which uses a fiberglass backing rather than paper (it's the paper that mold likes).

Note that these products aren't designed for wet applications, however. At best, they are designed for areas that 'are sometimes damp'. If you have mold issues under your sink, the problem is water. You have a leak somewhere. That has to be fixed. Replacing the sheetrock--even with one of these other products is just a bandaid and the problem will return if the leak isn't fixed.