This little guy was in a pot of three different cacti and I guess the other two were much more durable. Sometimes it's hard to read what your plant babies are trying to tell you. This guy started to get brown but was not mushy so I assumed he was getting sunburned and gave him more water. A couple days later he was squishy at the base and his little arms were coming off and I realized he was rotting. It's so sad when this happens and I feel like a failed mother but I always try to salvage plants when this happens.
First, dig up the entire plant, roots and all. It's important that you dig out all of the plant so that no nasty rot is left behind, especially if your cactus shares its home with others. Throw out whatever parts are brown and squishy. But save anything that still looks normal. Mine disconnected from the roots so I dug those up and got rid of them.
Now set aside your cuttings in a dry and sunny area. Outside somewhere covered or in a windowsill is ideal. You don't want any moisture getting on these guys. Let the cuttings sit for a few days until they develop a dry, solid callus over their cuts. Once they are completely dry you can reintroduce them to soil. No need to completely plant them, just let the tip of the plant slightly push into the soil. Do not water them for at least another week after this. If they decide to take again they will sprout roots and do their own thing. The best thing to do is leave them alone. I have had plants re-root in weeks and some did nothing for months and then suddenly planted itself again. It's pretty fascinating!