What would you rather pay for - birth control programs or 18 years of welfare? Poor people aren't going to stop having sex - and they reproduce faster than wealthier, more intelligent citizens because they are uneducated.
Protip - states with the lowest funding for planned parenthood, sex education and the like also have the highest rates of teen pregnancy, and are the most welfare dependent states in the nation. You want to save money? Empower women with easier access to planned parenthood, birth control, and fund the shit out of sex education.
Birth control is way cheaper than 18+ years of government assistance. I know which one I'd choose. Saying you want to get rid of both is just tilting at windmills. You're not going to stop these people from having sex. You need to educate them about it and make it possible for them to avoid conceiving when they are not capable of providing for a child.