Like many things, there are pros and cons. I don't feel that a simple 'yes' or 'no' really addresses the issue.
Municipal, county, and state regulation of various trades has been a fact of life for many years. Typically this is not seen as intrusive, and usually of benefit to the community.
Examples include barbers and hairdressers, skilled trades such as plumbers, electricians, and carpenters, private investigators, and public school teachers. All vary by jurisdiction, of course.
In some places, one may receive a 'license' by applying for it, perhaps submitting to a background check, and paying a fee. In some places, there are tests administered. In still others, private sanctioning bodies exist, which the government entity recognizes, and which members of said body are accepted and licensed by the government entity.
Professional licensing of the latter sort includes doctors, dentists, lawyers, veterinarians, and psychiatrists. It can include psychologists, chiropractors, marital and other types of counselors.
So licensing is not that unusual.
With regard to the pugilistic arts, most states license, regulate, and monitor professional boxing, and many are now beginning to accept professional MMA bouts as well. South Caroline is considering it now, Michigan has just accepted it, as well as a number of other states.
However, to the best of my knowledge, most states do not regulate the private teaching of martial arts. Anyone can hang out a shingle and teach whatever it is that they wish, even if they learned it out of a book or made it up out of whole cloth.
That leaves the risk for investigating such training centers strictly on the student, caveat emptor. Their sole remedy if they find themselves the recipients of faulty training - or worse - dangerous training - is limited to tort law (civil lawsuits).
Part of the problem with licensing, as I see it, is that there are no standard associations or organizations in the martial arts world. There are hundreds, if not thousands, of styles, and each of them feud and split off from each other with the regularity of certain Christian churches which will remain unnamed. Each lays claim to legitimacy and some deny that their other branches are valid - it is ugly.
Unlike, say, poker dealers, who are licensed, but they can take a test to prove they know their stuff - there is no one way to block a kick or punch, no standard anything, and therefore, no way to 'test' that an instructor can do what they say they can, let alone testing if they're capable of teaching it to others.
So why bother with licensing at all? It would be a nightmare, no doubt about it.
Well, one reason is because of the danger of physical injury. An ill-trained or ill-informed instructor can hurt students, or teach them unsafe training methods that could end in them hurting themselves. All well and good for adults, perhaps, who are supposed to know the risks going in, but for children, whose parents may simply assume that 'Sensei He-Man' knows what he is doing, after all, look at all those trophies and awards on the wall...
Here's a thought. How about licensing not the instructor, but the training facility - and treat it like a restaurant or tattoo parlor? That is, regulate it as if it were serving food or performing minor medical procedures - cleanliness, safety, and keeping track of injuries and requiring the training center to report any injuries to students during training. It doesn't prove that the instructor knows what they are teaching, but it might help to put the 'bad' MA centers out of business.
Anyway - in general, I'm not in favor of licensing and regulating things just to do it. If there is a problem, and if licensing and/or regulation provides a public benefit that can be defined, then I might be in favor of it. Frankly, I don't much care if barbers are licensed, because I cut my own hair. But I'd hate it if we didn't license electricians and I hired one to wire my new bonus room and it burned down a couple years later because he was an idiot and used the wrong guage wire, for example.
And with that, I'm off to the dojo.