Cleaning/refinishing advice for water-soaked window frame

My house was built in the 1950s, it has exactly 2 windows that have been updated (read: actually seal) since then. This is one of them. It is behind the kitchen sink. When deep cleaning it today, I realized that apparently the sink has been splashing behind the screen and has started to rot the wood. Since this is one of my "nice" windows, I want to fix this before it gets worse. My instinct is that I should sand it down, stain or seal it, and put caulk up against the glass part. Is that the correct way to handle this? What kind of varnish or what should I use after sanding it? I'm hoping to avoid needing to replace that part of the frame, because I'm sure there's more to it than just cutting, sanding, staining a new board. But if that's the only solution.... Please advise the best way to do that 😅