No, I do not believe there are many unfit mothers. And I am extremely wary of anyone thinking that they have some measuring stick to judge such a thing.
You seem to be specifically looking for an answer to what a mother should do if she is poor, uneducated, drug addicted, and on government welfare. How about instead of worrying about how we are going to to make sure the child doesn't follow in mom's footsteps, we start helping women in this situation through private associations. Don't focus on the baby. A baby is a person just like it's mom. Both need help and the best way to help the baby is to help the mom.
Are you looking for a way for the government to help? They can prosecute those who enslave and kill others through poisonous drugs and prostitution of poor girls (which may mostly be economically based rape.)
But doing the positive thing is up to us. Freely and individually. If it effects a family member then see to it that the mother has your support to better her circumstances. If not by supplying food or shelter, then at least by good will and wise, loving advice. Never labeling her as an unfit mother or her children as unwanted. Every woman has the ability to be a superb mother to her child. Every child is worthy of life and love.