If he is the father, he has every right to see his children and go for custody. If he is a recovering alcoholic, shouldn't that be a sign that he wants to sort his life out and put things right?
If you are married and not divorced he has exactly the same rights to see his children as you, unless a Judge has decreed otherwise. If he walks to your home and wants to see the kids, there's nothing you can do to stop him, legally. As your still husband and wife.
May I ask why is he applying for custody; and what the reason for the split was, as if it was infidelity he's pretty much lost everything.