Current State Laws, enacted before Minnesota was a state, make adultery illegal.
Minnesota Statute 609.36 says that,
When a married woman has sexual intercourse with a man other than her husband, whether married or not, both are guilty of adultery and may be sentenced to imprisonment for not more than one year or to payment of a fine of not more than $3,000, or both.
But the statute goes on to say that prosecutions will not be brought unless the husband or wife involved makes a complaint to the authorities. There's a limit of one year after the adultery for a prosecution to be brought.
And, the man involved isn't guilty if he was unaware of the marital status of the woman at the time.
And yes, it's just married women, not married men, who commit adultery, who are technically committing a crime. There's clearly an argument for removing this from the legislature for being unfair and obsolete, although the Minnesota Family Council thinks that the law should be made fair by expanding it to apply to married men too, believing that it will strengthen Minnesotan marriages.
Women and sex are the subject of another archaic law.
Minnesota Statute 609.34 says that,
When any man and single woman have sexual intercourse with each other, each is guilty of fornication, which is a misdemeanor.
So, no sex for single girls in Minnesota. It would seem that the only legal way for women to have sex in Minnesota is when married, with their husbands. And putting the two together, makes it illegal for men to have sex with single women, or women married to other men, which would just leave their own wives. So it's actually a little fairer than it looks, despite the fines and prison time provided for in the law concerning adultery.
In reality, Minnesota adultery laws are never enforced. In some states where adultery is illegal, their law may be used in divorce cases. But, unlike in some other states, Minnesota is a purely "no-fault" divorce state. That means that neither party has to prove fault or blame for a marriage's failure, and whether one or both spouses have committed adultery or not is irrelevant to the divorce proceedings.