A 2008 article by Anindya Sen (University of Waterloo, in Canada) and May Luong (Statistics Canada), published in the journal Contemporary Economic Policy (ungated here) provides some useful evidence. Sen and Luong used provincial data from Canada for the period from 1981 to 1999, and looked at the relationship between beer prices and gonorrhea and chlamydia rates. Interestingly, over that time beer prices had increased by 10%, while gonorrhea incidence had decreased by 93% and chlamydia incidence had decreased by 28%. They find that:
...higher beer prices are significantly correlated with lower gonorrhea and chlamydia rates and beer price elasticities within a range of -0.7 to -0.9.In other words, a one percent increase in beer prices is associated with a 0.7 to 0.9 percent decrease in gonorrhea and chlamydia rates. So, if the excise tax on beer increased, then the incidence rate of STIs would decrease. However, it is worth noting that the effect of a tax change will be much less than that implied by the elasticities above. According to Beer Canada, about half of the cost of a beer is excise tax (although that calculation is disputed, I'll use it because it is simple). So, a 1% increase in beer tax would increase the price of beer by 0.5%, halving the effect on STIs to a decrease of 0.35 to 0.45 percent. Still substantial.
Of course, that assumes that Sen and Luong's results are causal, which they aren't (although they do include some analysis based on an instrumental variables approach, which supports their results and has an interpretation that is closer to causality). However, in weighing up the optimal tax on alcohol, the impact on STI incidence is a valid consideration.