Absolutely. We need to change our culture like we did with smoking -- there was a time when almost everyone thought that smoking was "cool," but nowadays smoking is almost universally loathed by people who don't smoke.
Similarly, we need to make it a cultural norm that if you are sexually active, you are on birth control at all times, unless you are specifically trying to have a child. No exceptions. It would help if we could develop a birth control pill for men so that both sexes would share this responsibility.
I wouldn't go that far but I'm glad you like the idea of helping the less fortunate. I'm sure a pill for men could be developed without the government running up more debt. If someone doesn't want birth control then we shouldn't have to have anything to do with it. However, for low income women over the age of 18, we should be providing some form of limited assistance in order to prevent unwanted pregnancies and limit abortions.