If it was mine I would cut out whats dead, pot up the little ones so it dosen't get too crowded and see what happens. If its sprouting from the base its just going to grow back anyway.

Don't attempt to remove the new growths at the base just yet. You have a few choices here - you can leave the mainstem in place and let it sprout from the top if it looks like its going to, and you can leave the babies at the bottom to grow on and form a bushy area at the bottom. Or you can remove some of the babies, but you will need to wait a bit longer - each one must have some root with it when you remove them, so a sharp knife, cut downwards and make sure there's root attached. This will be more likely the longer you leave them.

Thanks a lot for this info. I will surely be passing this on to my fellow cordyline growers.