While many Democrats and women find Palin's support for criminalizing abortion to be outrageous, a recent survey by the Pew organization finds that roughly 40% of Americans generally agree with her views.
Combined, 41% of Americans answer that abortion should be outlawed in either in all cases (15%) or most cases (26%). Among White evangelicals, who Pew measures at roughly 30% of the American public, this support shoots up to a combined 62% (19% all cases, 43% most cases).
UPDATE: One problem in comparing this poll to Palin's specific views is the unclear nature of what counts as "most cases" for respondents. As commenters below note, Palin's position is to only allow abortion in cases where there is threat to the mother's life, but not for incest or rape. Few polls separate out the conditions differentiating between rape and incest and threat to the mother. Below are two additional examples from PollingReport.com.
Time magazine, August 2008
Which of these positions best represents your views about abortion? A woman should be able to get an abortion if she wants one, no matter what the reason, up until the time the fetus is viable, that is, can live on its own (46% support). Abortion should only be legal in certain circumstances, such as when a woman's health is endangered or when the pregnancy results from rape or incest (40%). Abortion should be illegal in all circumstances, even if the mother's life is in danger (10%)."
NBC News Sept. 28
"Which comes closest to your view on abortion: abortion should always be legal (25% support); should be legal most of the time (24% support); should be made illegal except in cases of rape , incest and to save the mother's life (37%); or abortion should be made illegal without any exceptions? (10%)"
Perhaps the best survey items on the specific conditions shaping support for abortion rights have been asked in polling sponsored by Fox News. In this case, we see that when the specific conditions of threat to a mother or circumstance of pregnancy is primed in survey respondents minds, support for abortion rights shifts to 70%.
FOX News/Opinion Dynamics Poll. Oct. 23-24, 2007.N=900 registered voters nationwide.
"Please tell me if you think abortion should be legal or illegal in each of the following situations..."
"If the pregnancy was the result of rape or incest"
70% Legal 21% Illegal 9% Not sure
"If the pregnancy puts the mother's life at risk"
73% Legal 15% Illegal 12% Unsure
"If the baby has a fatal birth defect"
53% Legal 30% Illegal 18% Unsure
"If the pregnancy is unwanted"
39% Legal 50% Illegal 11% Unsure