It would probably save tax dollars over time. The women that want abortions and need the gov't to pay for them would in turn be more likely to be a tax burden on the system anyway. If they can't pay for an abortion, pre-natal and obstetric care is much more expensive than that so we would be on the hook for 10 times the amount of the procedure right off the bat assuming an optimal outcome. This doesn't even begin to include future hardships that are eligible for even more public assistance like housing and food and continuing medicaid coverage.
Abortions don't count in overall stats as far as infant mortality either, and women not wanting to be pregnant are not in the category of tending to care about their pregnancies so our stats would suffer and so would the children, not to mention the costs skyrocketing. High risk pregnancies are expensive, so are neo-natal specialists. Neonatologists do four more years of school than regular doctors, the nurses are all specialized and paid more and NICUs are really really expensive to run.
How about the gov pays for tying tubes of young women on assistance? When and if she is able to PAY to have the proceedure reversed, she can. It would beat murdereng innocents.
As a matter of practicality, I doubt there are a lot of young women on assistance that are not already in the system because of a pregnancy or because they already have children. A single childless woman would have a hard time qualifying for any benefits. So would a single man for that matter. At any rate your suggestion smacks of eugenics and government control. You're still spending money but mandating/manipulating a social behavior. This is only a conservative issue because the "fundies" got into the Republican party and are now controlling it. This is not a classic conservative issue, in fact it is a classically unAmerican type of thought. And why is it always put on the woman? Let's have vasectomies for all single men that are not married instead, eh? Doesn't sound so rosy now does it?
I am a conservative, but think this is one way that is reversable that could be used to discourage lifelong gov assistance. As for the vasectomies, if they are on public assistance, the same thing should apply: staple the tubes, and let them pay to reverse it.