This just isn't true. When it comes to reproductive rights, women have all the power. The responsibility doesn't fall on the women, it falls squarely on the men. The choices are made by the woman. The reponsibility is given to the man.
1. If a woman has unprotected sex with a man and a pregnancy results, the woman can make a choice. Even after she has sex that results in conception she can choose to abort the baby. She can choose to have baby. She can choose to give the baby up for adoption.
2. If a man does the same thing, has unprotected sex with a woman resulting in a pregnancy. He has no choice. If the woman chooses to abort his baby, he has no say in the matter. If the woman chooses to have the baby, he has no say in the matter (and he is still financially and legally responsible).
3. If a man and a woman both wants custody of the child. 9 times out of 10, the woman gets it.
I agree with you that the issue of birth control is very sexist. But in this day and age, it is the woman who has all the power and the men who have the responsibility.