This is an image of a begonia x giganticum. Needless to say, it looks a little sad. It, along with a “sister” plant that I have that thankfully is still doing well, are my oldest house plants. I have had them since 1978 (yes, that’s 40 years–that’s not a typo).
This particular plant started to do this “thing” I will call it (for lack of the more precise technical term–it’s here that you realize that I have no background in horticulture) in mid-2017. I gave up on it in August of this year. I took cuttings, but I am not sure that the cuttings have survived.
If you look at the spot where the stem meets the leaf, it’s rotting. That’s the “thing.” I don’t know why it’s happening and clearly I don’t know how to stop it. I let the whole plant dry to the point of wilting several times and that didn’t help. If I had to guess, I would say this is some sort of bacterial of fungal wilt.
So that plant is now compost. I just hope my poor other specimen survives. There’s a lot of history in these plants!
Gardendaze 
In the wild, those stems would behave like rhizomes, developing roots where they touch the ground. They would be getting buried by foliar debris that falls down from above, so that they would be more inclined to develop adventitious roots. As they creep along and root as they go, they leave the old stems behind to rot away. Of course that does not help much for your houseplant that is not out in a forest. These stems look remarkably healthy though. I do not see the rot. Otherwise, I would recommend adding a bit of soil if the stems were elevated above the soil so that they can not root into it. A slight bit of soil would bring the level up to the stems so that they could root into it.
If by ‘stem’ you mean the petioles that suspend the leaves, that is a different issue. If they are rotting, that is some sort of mildew. It looks like that is what is happening to the leaves off to the right. I do not know what to recommend for mildew on houseplants, but lower humidity should help more than less moisture in the soil.
It very well could have been a humidity issue since it did start over the summer, although we do have central air. Still, all that does is lower the humidity a bit in the house, I know–it certainly doesn’t remove it entirely.
I do try to grow these–& in fact almost all my plants–on the drier side. I have actually killed cactus from too little water. But I know that more plants die from over-watering problems, so that, at least, is usually not my issue.
I will try adding some very light soil over the rhizomes of the remaining one. Generally I have great success in rooting these–but that is when they’re healthy. We’ll see.
Maybe I will also try leaf rooting as well. I made that work once.
Karla