Let me preface this by saying - I am not a member of the Religious Right. I’m a good liberal, like so many other Splendid Straight-Dopers. That said-
If you absolutely believe that it is your God-given mission on Earth to keep people from getting it on, wouldn’t you want to promote the HPV vaccine? As one of the other posters pointed out, most people probably have never heard of it - or at least, don’t understand that it basically acts as a cancer-STD. That’s some scary shit. So if I were John Q. Nutjob, I’d try to get a message like this in every newspaper in the country:
“The Coalition of Self-Righteous Nutjobs applauds the development of the HPV vaccine. HPV causes cervical cancer, which kills X number of women every year, and is primarily spread through sexual contact. We urge women in particular to get this vaccine - although, since the effectiveness of the vaccine is unproven, abstinence from sex before marriage remains the surest precaution against cervical cancer”.
With a message like that, the nutjobs could sound compassionate while still scaring women. Now, more women know HPV exists, and the brimstone nuts manage to spread FUD (Fear, Uncertainty and Doubt) about the vaccine without condemning it directly - the effectiveness is “unproven”, after all. So now women have something new to worry about, and if the nutjobs are “lucky” this sort of campaign can spark a run on what will presumably be, in the beginning, a relatively rare vaccine. The shortages, in turn, could cause additional women to become afraid - and from the Nutjob Right’s position, nothing looks better. Scaring women into doing what they want is what they’re all about, after all.
Or maybe there’s something I’m missing here. This is how I’d play it if I were a nutjob - but then again, nutjobs don’t think as clearly as I do, by definition. 
(Disclaimer: By “nutjob”, I mean the radical religious right, the people who believe in government-facilitated evangelicism. I have nothing against religious people of any stripe, so long as they don’t try to force their religious beliefs on my, my friends, or loved ones.)