Shutdown Over Planned Parenthood?

I understand that women's rights organizations that support Planned Parenthood are a critical part of the Democratic Party's base. But if, as Senate Majority Leader Harry Reid insists, the GOP desire to cut off funding for Planned Parenthood is the last outstanding issue in negotiations to avert a government shutdown, Reid and President Obama should agree to the cut-off in aid and keep the government going.
I say this not only because I have no special commitment to Planned Parenthood and have deep reservations about their work and their worldview. But, I have been shocked that Democrats point to the cut off in aid for Planned Parenthood as the central point in what they see as an attack on women, when the cuts in aid to programs that aid the poor seem to be much more anti-women and anti-children than not. And, the President and the Democrats can say to their base, "Look, you did not show up at the polls last year so we got a Republican House. There is only so much we can do." If there really are that many women in the U.S. who are so deeply committed to Planned Parenthood that they would rather see the government shut down then cut off aid to the group, the Dems should win in 2012 in a landslide. As it is, I think many Americans are ambivalent about Planned Parenthood and will not care that much if its funding is cut off.
Meanwhile, the GOP is saying that more is at issue than Planned Parenthood funding. We will know soon enough if they are lying. If the final result is a continuing resolution with about $38 billion in cuts, they are lying.