Absolutely - assuming, of course, it’s thoroughly tested and safe etc etc etc. The same goes for the HPV vaccine, or anything else, either for myself (I’ve had it), or my children should I and the-man-whom-I’ve-not-met-but-hopefully-will-some-day-who-is-my-husband be so blessed with them.
I totally understand concerns about the side effects of vaccines, and yes there is some evidence which
maybe questions the safety of the HPV vaccine. (Routinely questioning vaccinations is basically the fault of the guy who perpetrated the MMR-autism link fraud in the 1990s. That guy should be in prison, but anyway…)
Reading between the lines of some posters here - and more particularly having had even more explicit conversations with people offline, some appear to oppose this welcome medical breakthrough (the cervical cancer vaccine especially) on the grounds that it might encourage kids to think casual sex is suddenly a bit safer and more-ok.
Some of these kinds of battles (abortion!) are most definitely worth fighting, others are just a bit silly. Children might be denied a potentially life-saving vaccine just because it might sabotage the abstinence-only message (which now in this case seems to be relying on a fear of dying from cervical cancer to promote abstinence, which isn’t exactly grand in my view).
Vaccines work best when everyone gets them - the principle of ‘herd immunity’. HPV is the most common sexually transmitted infection. So, by vaccinating children before they are sexually active (promote abstinence all we like…it doesn’t mean every hormonal 16 year old will follow through on it 100% of the time), there is a hope of dramatically reducing the prevalence of it for the next generation.
Opposing an effective vaccine that could thousands of lives a year with the vacuous assertion that it might lead to promiscuity is inexcusable. About 60-70% of girls have (apparently) had sex by the time they’re 18. The need for a vaccine is clearly self evident. My (“socialist”, British

) view is there’s even a case for it to be mandatory: many parents might not know to ask for it, even if they’d want it, and besides, even if it’s available in theory but costs $200, it’s
not available to everyone. (By making it mandatory, it’s therefore accessible, because as far as I know most states/countries make religious exemptions for this kind of thing anyway). It’s also a teaching opportunity - a time for doctors to talk to adolescents about sexual health, and to explain the full range of risks by having non-monogamous relationships (by premaritial sex, by extension).
Of course there are parents who are reluctant to give their 9-year-old girl a vaccine against a sexually transmitted disease (it would make me uncomfortable too). But you do have to wonder…not just because I’d be astonished to find very many 9 year olds who know what a cervix is, but also because doctors are
not going to say “Congratulations! Now in a few years you can go and have unprotected sex with strangers!” (When I got a tetanus inoculation, I don’t remember the doctor telling me it was now safe to dance on rusty nails). Even if as one gets older one does understand what the vaccine is for, surely the messages sent by parents every day for years (for respect, responsibility, the boundaries of behaviour - and abstinence) - count for more than the implicit suggestions of a vaccine.
If it’s medically safe, everyone should get it, and the same goes for HIV, HepB, and anything else.