How much does birth control cost? I dunno, maybe $20 a month, maybe $60 a month? As a society, we pay for birth control either through insurance, or directly, so it doesn't really matter how it is paid for, we still pay for it. It may very well be cheaper to the tax payer to pay for birth control than to pay for abortions or unwanted pregnancies.
It's a matter of practicality. I could care less myself, it's not really a big deal one way or the other, but there is probably little harm in insurance covering birth control and possibly a slight cost advantage, so I can't see why anyone would oppose it, except on maybe religious grounds. Religion really doesn't matter, if one doesn't want to use birth control, then no one is forcing it upon them. All I know is that I would much prefer to pay a couple of bucks a month more in insurance cost to cover birth control for someone else, than to pay a few hundred dollars a month more in either insurance costs or taxes to cover someone elses abortions or unwanted pregnancies carried to full term.