Laissez-faire in sex leads to all kinds of expensive negative-sum signaling and other games.

Not completely sure they're actually negative sum. They might look like that from a purely materialistic perspective ("lotteries are bad because the expectation value of how much money you'll have if you play is less than if you don't play" -- it is, but that also applies to going to the cinema), but if you factor in Fun Theory aspects...

Why not crack down on those, which would lead to a clear improvement by any utilitarian metric? If it's OK for the government to ban smoking and other activities harmful for public health, why not extend such treatment to sexual activities that have obvious and drastic public health implications?

I can't think of a way to achieve that (without large costs/risks/drawbacks).