Sex side first. I don't see how procreation could ever really be a recreational activity, hence my not seeing it as an entitlement. (I'm assuming you're seeing sex and recreation as a negative right. Lemme know if you're wrong on that).

However, from the practical policy point of view, I think throwing money at the birth control side makes sense. I think it's safe to say that the amount of money people have available for birth control is not a primary or even a secondary limitation on the amount of sex they have. So the moral hazard problem here is relatively small. By comparison, the potential savings of preventing unwanted pregnancies are much higher. So I'd argue birth control is a good investment for society, even if it isn't a moral necessity.