The line typically does not get drawn at birth. The legal cutoff in most countries that allow abortion is around 20 weeks after conception (it varies by a few weeks depending on the country). Before then women are free to get elective abortions and many do, after that abortions are only permitted in special circumstances and are very rare. Wikipedia has a graph of abortion timing in the United States and other relevant information, including a poll showing that most Americans think that first trimester abortions should be legal and third trimester abortions should be illegal.
An early stage embryo/fetus does not have a functioning brain and is not viable outside the womb. It seems like it could be reasonable to draw the line during pregnancy, before either of those two things have happened. Since brain development and viability are gradual processes, people could argue for drawing the line at a later part of the process, so a cutoff during the first half of a pregnancy could be seen as a relatively safe, conservative choice.
Nobody has an abortion in the third trimester because they couldn't make up their mind earlier. These only comprise one-half of 1% of all abortions anyway, and amount to major surgery. These are all tragic situations where the baby was very much wanted, but either the mother's health wouldn't permit her to complete the pregnancy, or the ZEF was found to have a genetic anomaly that would have sentenced it to a short life of inconceivable suffering.
Ironically, outlawing abortion increases the number of late-term abortions, as women who otherwise would have had their abortion earlier must now either save up to travel somewhere where abortion is legal, or find a clandestine provider.
Some thoughts that I don't remember anyone expressing on LW.
First let's get this out of the way: life does not "begin at birth". As far as life can be said to "begin" anywhen, it begins at conception. Moreover, the child's intellectual abilities, self-awareness or similar qualities don't undergo any abrupt change at birth. It's just an arbitrary moment in the child's development. So it would seem that allowing killing kids only before they're born is illogical. What are the odds that your threshold for "personhood" coincides so well with the moment of birth? Could it be okay to kill kids up to 2 years old, say? CronoDAS voices this opinion here.
But there's another argument in favor of considering the moment of birth "special". Eliezer linked to a study showing that the degree of parental grief over a child's death, when plotted against the child's age, follows the same curve as the child's reproductive potential plotted against age. Now, the reproductive potential of an unborn kid depends on its chance of survival, and the moment of birth is special in this respect. In the ancestral environment many kids used to die at birth. And mothers died often too, which made their kids less likely to survive. An unborn kid is a creature that hasn't yet passed this big and sharply defined hurdle, so we instinctively discount our sympathy for its reproductive potential by a large factor without knowing why.
How much this should influence our modern attitudes toward abortion, if at all, is another question entirely. As medicine becomes better, kids and mothers become more likely to survive. So if our attitudes were allowed to drift toward a new evolutionary equilibrium which took account of technology, we'd come to hate abortions again (thx Morendil). But then again, the new evolutionary equilibrium is probably a very nasty system of values that no one in their right mind would embrace now (won't spell it out, use your imagination).
Ultimately your morality is up to you and the little voices in your head. You think womens' rights trump kids' rights or the other way round, okay. But if you use factual arguments, try to make sure they are correct.
ETA: see DanArmak's and Sniffnoy's comments for simpler explanations. Taken together, they sound more convincing to me than my own idea.