A cure for all STD's, no.
For HIV, consider that if diagnosed before the virus has progressed, there are now multiple medications which, when taken rigorously, reduce the viral load via the standard blood test method to undetectable in an infected person, which is what it would be in a non-infected person. So, define what you mean by a "cure" - if you mean some way in which people infected with HIV won't automatically proceed to AIDS and die, a cure is here now and has been for some years.
I am sure I will be attacked for this comment. I'm not meaning to imply HIV is over or is nothing to worry about - it's by no means over and remains something to worry about. People need to be tested and to practice safe sex. Getting an HIV diagnosis can be devastating, the meds are expensive, some have negative side effects, there is amazingly negative stigma in the world about HIV, etc etc etc.
In a way though, doesn't it make more sense to spend more of that drug research money in prevention and testing efforts?