STD's aren't a compelling enough interest to regulate a fundamental liberty. Consensual sex between adults has been considered a fundamental liberty covered by the substance of the Due Process clause of the Fourteenth Amendment.
Your point has some logic behind it, but as a legal argument, it is a non-starter.
Any activity which causes a detrimental effect on society at large is a legimate target of reasonable regulation even under the strictest scrutiny. The right of consensual sex has been historically limited by every legislature in this country. Even Nevada has regulations regarding prostitution that are designed to protect its citizens from the contracting of STD's and street walkers in Nevada are often treated much worse than street walkers in Nebraska.
AIDS costs the country hundreds of millions of dollars per year. Illegitimate childbirth costs the country billions of dollars per year.
As long as someone's freedom infringes upon the rights of others, in this case the right not to have to pay the consequences of illicit sexual activity, then those whose rights are impacted have as much right to regulate that activity as those who engage in it to practice it.
The 14th amendment was never designed to protect the right of homosexuals to receive the benefits of marriage as homosexual marriage has absolutely no postive impact on society, whereas heterosexual marriage does and always has had a positive rather than a negative impact on society.
This twisting of the words of the constitution to mean things they were never intended to mean undermines the whole principle of constitutional government. We might as well tear the document to shreds and run the country based on what we all think feels right.
If you think the 14th amendment in any way prohibits the states from regulating sexual activity, then I dare say that you have been brainwashed by your law school professors. Don't believe anything any of them tell you. Keep your mind open and your Christian principles forefront. They are no smarter than you, indeed they are trying to make you as stupid as them. Don't let them do it.