Oh poor mama plant!
If it were my plant I will have to remove Mama from the babies, split them, make sure you are wearing gloves, then sprinkle some cinnamon on the ends of the remaining good ones, just to protect it from the rotting. As for the mama plant, I would separate it from the rest, and since the rotting is coming from the bottom, you can try cutting it up, remove the rotten part and throw it out. Then allow the remaining fresh part to callus and dry, you can also sprinkle some cinnamon at the cut end. Not sure if it has enough energy to heal, but you can try to save it.
Were you growing this outdoors? If it were outdoors and it got cold and rained on, that is what happens. Now if it is indoors and in too much shade and getting more water than needed, it will do the same. It really likes lots of warmth and light. During summer when it is really very warm outdoors it can take the rain to a certain point, provided drainage is very good. But as seasons change got to protect it from cold. If indoors, give it as much sun, a south facing window will be perfect for it.
I have a Euphorbia that has been munched on, exposed to cold and rain and it survived, as long as I was able to let it dry and heal.
