Many people think we are bound by so many rules that we cannot do anything unless the church "approves" it. Although we have direction from the scriptures, our Prophets, and church teachings for many important things in life, for the most part, we are expected to make our own choices for the well-being of our families and loved ones. 
 
Your friend is incorrect in her assumption that she should stay with this guy. If the life or well-being of your friend and her child are in jeopardy with him around, she should not be with him.  
 
Forgiving a man like this DOES NOT mean you have to stay with him.

Though Christ said 'Forgive Everyone,' that doesn't bind you to stay in a situation you don't want to be in. 
 
Just because someone "served their time" in prison for child molestation, and I forgive them for what they have done, doesn't mean that I should have them babysit my kids.  
 
Again, No. It is not in the Church's teachings that she would have to stay with a man if she didn't want to.

Forgive, yes. Stay with, no. *points to Phillip* He's got it, but I'm going to add my two cents. 
 
We are taught to "love thy enemies, do good to them that hate you," etc etc. So she should try to be kind to this man. However, that does not mean she needs to stay in a situation that is harmful to her and her child. 
 
She probably should not have had a child before marriage. That was a bad choice, but it's done now. The Church teaches that the best thing to do in these situations is to marry the father and try to make the marriage work. I personally would never leave my child anywhere near someone who drinks, smokes, or does drugs. The way I see it, she has two options. 
 
1) Marry the guy and do her best. She'll want to put her foot down on the alchohol and stuff, help him break that habit. I sure hope she's strong, because she'll need all her strength. I don't like this option, but there it is. 
 
2) Get as far away from him as you can. Find another man- someone kind, loving, and who she can love. Marry him, so the child doesn't grow up without a father. Or just try her hand at being a single mom. Or something that does not involve the child's original father. Putting the child up for adoption is a harsh choice, but it's also available if she can find a loving family. 
 
Either way, she should talk to her bishop and ask his advice. Explain the entire situation to him, and get his blessing for her and her child before making a decision. Hope this helps.