It is not clear to me that it is illegal. If Coursera has no presence in Minnesota, there are a lot of things that Minnesota cannot require of Coursera. For instance Minnesota cannot require the withholding and payment of sales taxes.
Therefore I'm not at all sure that Minnesota has the right to enforce these regulations. They probably can require that Minnesota residents not do business with Coursera. They can definitely say that Coursera courses are not worth college credits in Minnesota. But I'm doubtful that they can legally require that Coursera assist them one bit in their attempts to enforce those rules.
Not even close; Minnesota isn't saying that Coursera cannot operate in Minnesota, just that it cannot do so without approval -- just like how you don't have the right to have a huge demonstration without proper permits. There is well-established precedent that it is acceptable to have minor hurdles in the expression of free speech in the interest of maintaining an orderly society.
Edit: also, the problem is that they are calling it a "college course", this is a simple consumer protection issue..
My impression is that they can get away with requiring permits for demonstrations because there are physical space considerations, public safety, etc. Imagine if a state decided you needed a permit to write for a political blog. Same content as a protest may have, but without the meatspace messiness.
Of course, I did not mean to imply equivalence. I am only pointing out that restrictions to freedom of speech being acceptable for physical protests does not necessarily imply anything about what may be acceptable limitations in other scenarios.
