It's the door of the bathroom in the basement, which hasn't been used for at least one year. The relative humidity is 50%. I'm not sure how the rot happened, since it was before I moved in.

As I can find online, the typical way to repair is to either replace part of the jamb with new wood or to fill with epoxy after removing the rotten wood. However, my case seems not that severe, are the above still the right ways to go? If not, what should I do? Thank you.

Edit: The rotted part is as hard as the normal part of the door casing. I scrapped the loose paint, and also removed the tile in the second photo, and photos are added. I have tested with tissue paper and confirmed the door casing is dry. It's the bathroom in the basement, close to the exterior foundation wall. The shower door sweep is missing, I guess water came out from the gap when it was in use.