Because they can be set at a level high enough to discourage the action somewhat, but low enough that clandestine abortions are not chosen instead.
For example, where I live cigarettes are taxed at punitive rates (to the right of the Laffer maximum) in order to discourage smoking. It is theoretically possible to evade these taxes, e.g. by growing your own tobacco or smuggling, but taxes are not so high that this is an appealing alternative, given the risks and penalties for breaking the law. So we get less smoking than otherwise, and revenue for the Exchequer, but we do not have smokers turning to dangerous contraband cigarettes instead. It is possible to mess this up; some years ago, cigarette taxes were too high, and we had a problem with smuggling. But I think the principle is clear.
The same logic applies to a tax on abortions. Suppose that the risks and dangers of an illegal abortion are worth -$5000 to the pregnant woman. Then if we forbid abortions, any woman who values an abortion at over $5000 will have one in a clandestine manner. But if instead we put a $3000 tax on abortions, then the pregnant woman who values the abortion at over $3000 will have one legally and safely, and the pregnant woman who values the abortion at less than $3000 will not have an abortion at all.
(Numbers are for illustrative purposes only).
Now, no-one is proposing a tax on abortions. I can see why you would oppose the tax if you are stringently pro-life, as you would see it as facilitating murder. I can see why you would oppose the tax if you are stringently pro-choice, as you might see it as an impermissible restriction on personal freedom. But consider the group who think that abortion is immoral, and thus presumably ought to be discouraged, but claim that their objection to a ban is that such a law would have bad practical effects. Surely then they should be searching high and low for ideas for other laws which would discourage abortion with better practical effects - like a tax. But instead people advance the argument that banning abortion would be counterproductive - and then leave it there.
My conclusion is that this argument an example of status quo bias, and fallacy of the excluded middle. I think it is popular because it allows the arguer to seem "onside" to both pro-life and pro-choice interlocutors.
(Apart from the fact that taxing abortions would discourage poor people from aborting more than it would rich people, which is not exactly what most people would want -- unless the tax scales with the woman's income, but that would just give Yet Another incentive for people to underreport their incomes) maybe that just doesn't occur to people? Seriously, some of these homo hypocritus discussions badly fail Hanlon's Razor (i.e. forget that 84% of the population has IQs below 115, or forget the way people with IQs below 115 think). My grandpa complains if I ...
Or is the convention against discussing politics here silly?
I propose a test. I'm going to try to lay down some rules on voting on comments for the test here (not that I can force anybody to abide by them):
1.) Top-level comments should introduce arguments (or ridicule me and/or this test); responses should be responses to those arguments.
2.) Upvote and downvote based on whether or not you find an argument convincing in the context in which it was raised. This means if it's a good argument against the argument it is responding to, not whether or not there's a good/obvious counterargument to it; if you have a good counterargument, raise it. If it's a convincing argument, and the counterargument is also convincing, upvote both. If both arguments are unconvincing, downvote both.
3.) Try not to downvote particular comments excessively, if they're legitimate lines of argument. A faulty line of argument provides opportunity for rebuttal, and so for our test has value even then; that is, I want some faulty lines of argument here. If you disagree, please downvote me, instead of the faulty comments, because this post is what you want less of, not those comments. This necessarily implies, for balance, that we not excessively upvote comments. I'd suggest fairly arbitrary limits of 3/-3?
Edit: 4.) A single argument per comment would be ideal; as MixedNuts points out here, it's otherwise hard to distinguish between one good and one bad argument, which makes the upvoting/downvoting difficult to evaluate. (My apologies about missing this, folks.)
I'm going to try really hard not to get personally involved, except to lay down a leading comment posing an argument against abortion, a position I don't hold, for the record. The core of the argument isn't disingenuous, and I hold that this argument is true, it just doesn't lead to my opposing abortion. I do not hold the moral axiom by which I extend the basic argument to argue against abortion, however; I'm playing the devil's advocate to try to help me from getting sucked into the argument while providing an initial point of discussion.
Which leads me to the next point: If you see a hole in an argument, even if it's an argument for a perspective you agree with, poke through it. The goal is to see whether we can have a constructive political argument here.
The fact that this is a test, and known to be a test, means this isn't a blind study. Uh, try to act as if you're not being tested?
After it's gone on a little while, if this post hasn't been hopelessly downvoted and ridiculed (and thus the premise and test discarded as undesirable to begin with), we can put up a poll to see whether people found the political debates helpful, not helpful, and so on.