Well like I mentioned in an earlier post. If all there is, is a lightly enforced misdemeanor with say a few hundred dollar fine (or less!), I think it would deter very few people to stop committing adultery.

Your original question was "should adultery be illegal."
Well the correct answer is, in most states it already is. It just typically very lightly enforced. Back to the jaywalking example, I would say nearly everyone at some point in their lives jay walked across a street, it just so rarely enforced, it is barely a law, barely illegal defanged by light penalty and even weaker enforcement.

Another example: marijuana is illegal by federal law. But if you go to places like Colorado or Washington State, and check into a reasonably priced hotel, near a major airport or highway, I can almost guarantee you, (based upon my own experience,) you are going to smell pot. Marijuana is illegal by federal law, but no one cares, because it is not enforced. The term "illegal" is almost meaningless.

Adultery is already illegal in most states but it still occurs all the time.

Also, be reminded that in some parts of the world, adultery does come with very steep repercussions, like many years in jail or death.
I am not to worried about the burden of police, if no one expects a step up in the enforcement of the law. But I am very concerned if they were to step up enforcement on the burden to police, courts, jails etc, aka the entire already badly overburdened criminal/civil justice system.

If you are okay with the idea that adultery is illegal, but that the fact it is illegal is "irrelevant" then yes, it is also irrelevant if someone can prove cheating/adultery or not.

But if the already in the books laws of adultery being illegal was actually to mean something, then yes, being able to prove in the court of law cheating does become very important.