Please, try to answer without passing too much judgement on me.
I am prenant. The father and I were in a relationship at the time it happened, but recently (around 2 months ago) the father relapsed. He has been in recovery for a long time, but can't seem to stay sober for more than 2 or 3 months at a time, or stay faithful in the relationship. I would absolutely love my unborn son to have a father, but I'm terrified of what could happen. This man has a good heart, but as for being mature enough to be a father, he's just not there. Obviously, it was poor planning and stupid decisions on my part to begin with, but I'm in the situation nonetheless.

Do I put the father on the birth certificate, if I'm fairly certain he is going to have no part in this child's life? I'm scared that by putting his name on it, he will try to get custody of my baby. I truly believe that he is not capable of taking care of a child, and I'm scared to put my baby at risk. What do I legally have to do, or if it doesn't matter, then what should I do?

While you don't have to, it does not take away any of the bio fathers rights, He will still have the same rights to file for joint custody if he wants, he can still file for visits, you will still have to ask his permission if there is a new partner in the future who wants to adopt.
The only difference he will have to file for a DNA test to prove paternity first. ( and you may have to pay 1/2)

Also you not be getting any child support, which is a right the child has to receive.