Priests for Life has a wonderful resource for all those who are wondering about this question:
Elections Have Consequences: Does it Matter if the President is Pro-life? [htm] or [pdf]
In particular, we have good records of what our current President has done:
The Pro-Life Record of the George W. Bush Administration (Catholic Exchange: Bush's first term only)
President George W. Bush's Record on Life (NRLC: less detailed but covers both terms)
I think his Blanket Pro-Life Veto Warning was also very helpful in stopping anti-life legislation and in protecting pro-life "riders" such as the Hyde Amendment.
Abortions have dropped 25% from their peak of nearly 1.6 million in 1990; in 2005, the latest year for which figures are available, there were about 1.2 million abortions. That's nearly 400,000 fewer babies killed each year! Major reasons for this significant drop are:
1) State level pro-life legislation such as parental notification or consent, informed consent (women's right to know), waiting period, etc.
2) An improved Supreme Court that upheld these types of regulations.
3) A significant increase in support of pro-life views by Americans, due to education by pro-life churches and groups. The campaign against partial-birth abortion is thought to be a significant part of this. This extends also to doctors: an increasing fraction of OB-GYNs is avoiding the abortion business.
The drop was definitely not due to primarily economic reasons (I will post my analysis on this soon), nor to increased availability of contraception (USCCB Pro-Life secretariat's Fact Sheet: Greater Access to Contraception Does Not Reduce Abortions).
Now this is all relevant because, as I mentioned earlier, Obama has promised to sign FOCA: "Well, the first thing I’d do as president is, is sign the Freedom of Choice Act." - from PPAF address cited above
As the USCCB Pro-Life media campaign notes: "You Can't Reduce Abortions by Promoting Abortions"
Radical abortion rights groups and their allies in Congress are promoting the extreme pro-abortion agenda of the so-called “Freedom of Choice Act” (FOCA). If they have their way, reasonable, widely supported and constitutionally sound abortion regulations will be knocked down nationwide. Unlimited abortion-on-demand will become our national policy.Also see our post on Cardinal Rigali's recent statements about FOCA.
And the abortion rate will go up, not down.
If we're going to ask what a President can do to reduce abortions, we should also ask what he can do to increase them. And Obama has already committed to promoting abortion through FOCA, which will surely increase them.
Update (10/15): The USCCB Pro-Life Secretariat's Richard Doerflinger offers more insight into abortion increases under FOCA.