Because it makes the perfect wedge issue is why.
Or, to be more thorough, this is something I've been thinking about ever since anti-choicers managed to get Komen to join in their crusade of shunning Planned Parenthood for what I've deemed below-the-belt health care.* The "breasts good/vaginas icky" divide that rules a lot of wingnut thinking on this is typical of why they obsess over sex. Sex and gender provide lots of opportunities to label people and divide them into categories. This in and of itself isn't bad. Labels are useful things that can clarify, as long as they're kept in their place. But conservatives---being black and white thinkers---see different groups labeled and imagine they must be total opposites. And that they therefore must be constantly at war with each other.
And boy do they like that! Nothing works better for conservatism than encouraging people to exaggerate their differences, minimize their similiarities and then imagine themselves as in constant struggle with each other. While we ordinary people are fighting each other over our supposed differences, we're not looking working together on issues that matter to all of us. There are a lot of places where differences to fight over can be found, but sex and gender strike so close to home for people that it's a well of opportunities for divide-and-conquer the right just drinks from constantly and frankly instinctually. You can probably think of the most prominent examples. There's the What's the Matter with Kansas? issue, where working class white people are encouraged to obsess over the dirty sluts that are supposed so different from them, and to bring us back in line, and this obsession keeps them voting Republican even as doing so hurts their pocketbook. Of course, the attempts to make straight people think LBGTQ people are foreign and subversive and strange is another example, and luckily one where the public is slowly beginning to stop buying what conservatives are selling.
Of course, the most consistently important artificial divide conservatives want to stoke, at least when it comes to sex, is between men and women. The idea that men and women might realize that we're not, in fact, complete opposites and that in fact our interests overlap the vast majority of the time terrifies conservatives. They need men and women to see ourselves as enemies who constantly struggle over power and sex, and definitely don't want us to look at each other as friends. That's one reason there's so much hostility with regards to reproductive rights. It's critical to conservatives ideology to imagine sex as a source of power struggle between men and women that prevents equality and friendship. They want a world where men are always trying to get it, and women are always trying to avoid having to have it, which makes platonic friendship impossible (because of suspicions that he's trying to get one over on you) and makes sexual relationships fraught. Reproductive health care changes that completely, giving women an opportunity to explore our desires without worrying too much about unwanted childbirth or disease, and when we do so, we realize that actually, we're just like men. And now instead of sex being the source of friction between men and women, it can bring people together. (Also, platonic friendships are a lot easier because people who are sexually satisfied aren't injecting sex into every interaction.) There's a reason that the same people who violently insist that men and women are complete opposites cringe at the idea of women having full reproductive health care; they fear men and women learning that in fact, we're basically the same. Nothing distracts from how alike men and women are like making women subservient to our biology while men are allowed to be free.
Feminism is dangerous not just because it can bring men and women together as allies. If anything, conservatives are just as terrified at how feminists have indicated that women are oppressed as a group and has encouraged women to stand together to resist that oppression. Turning women on each other is a major part of reactionary politics, and sex is a great weapon to use for that purpose. Women are still treated like the sex class, and that creates a lot of opportunities to divide women in various ways according to perceived sexual behaviors and desirability. There's a reason that conservatives are starting a "OMG GIRL SCOUTS ARE DIRTY SLUTS" campaign; it's about dividing women into "good girls" and "bad girls" and pitting them against each other, even before they even really have any kind of sexual urges at all. I think that's one reason they went after Komen so doggedly. Breast cancer is considered, inaccurately, as more of an older woman's concern. You start your mammograms just as your fertile years are winding down. Below-the-belt health care is considered, inaccurately, more of a younger woman's concern.** It's clear to me that they see this as another divide-and-conquer strategy, trying pit these two groups against each other. For antis, especially, they want to get more older women to snarl at younger ones that we shouldn't be clamoring for health care resources, but instead should just keep our legs shut.
This is bullshit, and I suspect won't work as well on older women as anti-choicers might think. My experience tells me that very few women go through menopause and then turn on their younger sisters who still have the needs of fertile women. On the contrary, I tend to find that older women are often more thoughtful and nuanced, having had enough experience to see that the division between "good girls" and "bad girls" is a lie, and that we're all bad girls. Sure, there are plenty of middle-aged and eldery church ladies, wagging their fingers at the sluts, but I suspect most of those women have been church ladies their entire lives, and have always preferred to throw stones at other women than see that we're all in this together. But since finger-wagging church ladies are a huge portion of the anti-choice movement, I could see how they'd see this as a way to beef up the divide between younger and older women. But even though I don't think this will work as well as they might hope, it's good to be mindful of it. And ask ourselves when we're getting all bent out of shape about the nerve of some people being at a different place in life and having different sexual and health needs, we have to ask why are we letting ourselves get riled up like that?
*Seriously, how long is it before fundie women start bragging about how they don't go to filthy gynecologists at all? On the fringes, you already have Christian fundies who believe that god wants women to give birth at home with a midwife, so I guess they're already there, looking for a way for women to go their whole lives without having a doctor address below-the-belt care. With the extremist turn as of late, I won't be surprised to see these sentiments spreading. Anyway, you can read my piece about the above/below divide wingnuts are carving out, and how to hold corporations who are coddling this accountable.
**I'm not an expert by any means on breast cancer, but I do know that the assumption that you don't need protection after you go through menopause means a lot of people being diagnosed with STDs when they thought they were too old for that.