Rain comes down the mast. There should be a drain hole in the aft end of the glass mast bucket. It gets clogged. Also, there should be a little dip or channel on the aft end of the aluminum mast step to let water drain out there. It also gets clogged. The mast slips over the mast step and unfortunately the two aluminum pieces will corrode and sort or weld the mast to the step. I have had problems with this in the past. The last time I pulled the mast I cleaned up both surfaces and smeared each with Tefgel before we restepped the mast.
Which year and model is your Morgan? The earlier models had the mast "bucket" as Terry talks about. Many of the newer models, 383 & 384 did not have such bucket. If yours has the bucket then look for the drain holes as Terry explains. Without the bucket you will have to explore to find the system that drains away the water from the mast. There surely will be a method. Maybe an owner without the bucket can help with that.
Glassed in tubes for limber holes that lead to the bilge on my 383, Jim. The tubes didn't go below the bottom of the mast area so it held a cup of water. Smallish drain hole in the mast and the aluminum step that it sits on. It would clog up with any dirt & debris (bugs, etc). So it would retain rain water in the mast and "ooze" moisture.
I leave a 4-5" piece of stainless steel wire just sitting in the mast bucket. Two or three times a season I reach in and use that wire to clear the debris from the weep hole at the base of the mast. The hole in the bucket that allows the water to drain out to the bilge over the keel sometimes get clogged up and is easy to clear out.
I don't know where along the line the bucket disappeared. My 382 #53 has one. By your photos, the drain hole in the mast and the drainage slot in the mast base is the same. Looks like the only difference is the bucket has a hole that allows the water to reach the bilge on top of the encapsulated keel and below the fuel tank. Does that sound right to you?