It is everyone's right to have or not have children. So no, birth control should not be mandatory for those that can't afford or are unfit to be parents. However, it should be mandatory that all parents be solely held responsible for providing basic needs (food, shelter and healthcare) for their children (without gov't help). If they don't, gov't should step in, garnish the parents' paycheck and provide for the children. If they are unfit, then CPS should step in and again the parents' paycheck should be garnish to provide for the children.
Our nation was founded on liberty. Making birth control mandatory is too intrusive.
However, some people shouldn't be parents. That being said giving birth does NOT mean you must raise the child. People who show a history of neglect or abuse should be put under a court order where if they have another child the child is immediately taken after birth and placed in a foster home or with an adoptive family. This would strongly encourage birth control or abortion but in case they don't then the child would still be rescued from being raised by an unfit parent. This way we protect the child but don't intrude onto what a person does with their own body.
But this shouldn't be based on income. Poor people for the most part are trying to make more money, but the economy isn't always fair or perfect. The free market is great but there needs to be a social safety net to help those who fall on hard times, especially parents.