From the various researches found, we can draw the conclusion that
Yes, it does decrease abortion rate in the place where it is banned
Education, policies clearly influence
We can find a lot of work regarding the influence of good sexual education explaining lower abortion rate in some countries such as Netherlands. It is truly intuitive to believe it, but research demonstrate it. In 2003, a study over abortion data of the United States from 1974-1988 induced that abortion policies influenced greatly the rate. It concluded that:
Restrictions on Medicaid funding for abortion are correlated with lower abortion rates in-state and higher rates among nearby states. [...] A larger number of abortion providers in a state increases the abortionrate, primarily through inducing cross-state travel.
However, it does not directly answer your question since it does not imply the state banned abortion.
Legalization and its effect on the long term
We can find a lot of research notes and papers on the abortion rate trend in the various countries of the world. All of them draw the conclusion that the trend is downward in the majority. In particular, some draw that conclusion:
Legalization of abortion and access to abortion services do not lead to increased reliance on abortion for fertility control in the long term; in developed countries with these conditions, the predominant trend in abortion rates has been downward.
So at least, we know that legalization does not spread the use of abortion as a contraceptive method if alternatives exist.
Effect of the ban in United state
As we said, the legalization does not increase the number of women that want to rely on abortion at a certain moment of their life (at least over the long term). That said, we could conclude that for similar education (on contraception, religion), the rate should more or less the same inside a country. The book "Abortion Rates in the United States: The Influence of Opinion and Policy" study the various researches published on the subject to draw few conclusion. First, public opinion is the most influential factor between state were it is legal. Second, reported abortion number were much lower before legal abortion. Third, illegal abortion increase dramatically the chance of dying.
There is no clear statement or proof that illegal abortion result in lower abortion rate (since we cannot count unreported abortion). But from known statistics, from the facts on influential factors we can draw the conclusion that yes it decreases the abortion rate in the place were it is illegal. However, as stated in the beginning, it might artificially increase the abortion rate in the country or state beside where it is legal.
Hope that my little literature review has helped.