First of all, please realize that you are using the wrong word when you say “unconstitutional.” It is only actions and laws made by the government that can be unconstitutional. As a private citizen, you cannot do anything that is unconstitutional. You can do things that are illegal, but not things that are unconstitutional. So, what I think you are asking is whether the government could make laws against certain sexual practices that could threaten public health. In other words, could we make such practices illegal?
Such laws might be constitutional if they were A) truly made for health purposes and B) enforced against everyone. Otherwise, they would probably be violations of the equal protection clause of the 14th Amendment. So, for example, let us say that you wanted to ban anal sex because it would be more likely to spread HIV. The government would have to show that the ban was really for the sake of health purposes and, most importantly, it would have to be sure to enforce the ban against both same-sex and opposite-sex couples. If it did not, courts would probably rule that the law was simply a way to ban homosexual conduct without coming out and saying that is what you are doing.
Depending on the court, even that law might not be upheld. A court might say that, while protecting public health is a legitimate goal for the government, banning all anal sex unnecessarily infringes on our rights as adults. A court might say that there are ways to protect public health that would not intrude on our rights as much as the law I describe above.
Outside of the constitutional problems, it also seems unlikely that such laws would pass a legislature today. This is mainly because such a law would essentially be unenforceable. To enforce it, police would have to go into people’s houses and watch their sexual behavior to make sure they weren’t violating the law. You might charge someone with such an offense if they were caught engaging in this kind of conduct in public, but having sex of any sort in public is already illegal. So, unless we are willing to have the police enter people’s homes to regulate their sexual conduct, it would be impossible to enforce such a law.
Overall, then, it seems very unlikely that a law banning certain kinds of sexual conduct because they are a danger to public health could be passed and found constitutional in today’s society. If we were truly going to make a law like this to protect public health, it would be unenforceable. If we were going to pass the law as a roundabout way to ban homosexual conduct, it would be unconstitutional because of the Supreme Court’s ruling, in Lawrence v. Texas, that the government cannot ban homosexual conduct. For these reasons, I do not think that we could make certain kinds of sexual behaviors illegal because of their public health implications.
The above is a great analysis of the issues involved in this particular question, but you might be interested in what is a mostly historic sidelight. There was a time, in some states as recently as 1980, when states required blood testing for sexually transmitted diseases (STDs) as a condition of marriage, particularly for syphilis. There was routinely a three-day waiting period before a couple could marry, to await the results of the blood test. Right now, there are only two jurisdictions requiring blood testing for STDs. However, there is an effort to put this requirement back on the books in Oklahoma, and I suspect this idea could gain momentum in other states as a consequence of the Supreme Court's recent ruling on gay marriage, particularly testing for HIV. The justification for this would be public health, to be sure, and I suppose it's possible that this could happen again, but there are serious privacy consequences that might outweigh a concern for public health. Certainly, the government can no longer regulate what two consenting adults do with one another sexually within the privacy of their own home, which was decided in a case called Lawrence v. Texas in 2003.