That would be a preferred solution. Eliminating government welfare entirely would be the best solution. If a woman faced the prospect of having to support her own kid by the sweat of her own brow, working at Burger King or wherever, they would pay much more attention to their birth control methods.
Two factors would work against that, though:
1) The tendency of welfare moms to put their own comfort ahead of their baby's. We would keep seeing news stories of children being malnourished because of "heartless" caps on welfare payments.
2) Middle-class women would vote against it. If a poor woman does not have welfare as an option, one of her other options is to compete with working-class and middle-class women for the affections of guys with decent jobs, whether as wife or mistress. What welfare currently does, is take millions of young women out of the competition for the pool of desirable men, and there are lots of middle-class women who would want that to stay that way.
Very interesting point that I have never considered.