That is a common problem unfortunately. Moisture often enters from two points. The ends at the bottom of the glass seal, and from the bottom of the door where it contacts the track. Another problem is the actual wood used to make these doors, it is young timber which in simple terms, sucks! All the old timber is long gone unfortunately.
Regarding the repair. I use these two products. The first product seals and hardens the wood. Wood Hardner.
The second product is the filler. Filler
. You can also use carpenters wood filler which is not as strong but does accept stain a little better.
What I would do is hog out the rotted wood, and sand the area. Put painters tape forming a border around the rotted area, go a little higher and wider and apply the hardner being careful not to get too much on the tape. Once it starts to set remove the tape and allow it to dry. Once dry apply the filler, allow it to harden and refill as needed, overlapping slightly over the hardner. The reason for that is the hardner doesn't accept stain. One dry sand and clean it, get a stain one to two shades lighter than the stain you have, test it and stain the area. The filler tends to stain darker, so you might have to play around with it. When that dries scuff sand the door and apply two coats of Marine Spar Varnish in the finish you like, satin or high gloss. Taking the door down is always better but, you be the judge.
Another way is to actually remove the bottom section, carefully. Use it for a template, make a new section using the old gaskets and seals. That is not an easy job, and you'd need some nice woodworking tools, patience and woodworking skills. Either way good luck!
If you have any questions PM me if you'd like.