If the style is a sport fighting style, if they and their students win, its a good sign.
If the dojo or dojang or practice hall or whatever it is called in the relevant style is in a mall, or has nothing but children practicing in what is essentially an athletic day care, that is a bad sign.
If the dojo is part of a chain, that is a bad sign.
If the dojo has contracts, and a big focus on money, and slick advertising, that is a bad sign.
Otherwise, unless you are a senior martial artist already, you cannot judge the technical proficiency of the teacher--you haven't yet developed the eye and so you cannot judge them technically.
You can find people you trust who are senior level martial artists, if you know any. Here I am talking about 20-year people. Their opinion can help guide you.
You must ask your self why you want to start practicing a martial art. There are many legitimate reasons:
- Fun, social activity with some physical activity
- Sport and competition
- Self defense
- Self improvement in the zen sense
Technical proficiency of the instructor is only important for sport (where you can see the winning or losing of the students, and the trophies on the wall) and for self-defense.
There is a classic martial arts joke that goes like this:
A guy in New York gets mugged, and the mugger beats him up and takes all his cash.
He swears this will never happen to him again, so he goes to his local dojo, and
tells the instructor he wants to learn to defend himself so that he won't get beat up
again. The instructor says, "Well, you live in New York. You might get beat up
and mugged once every ten years. If you join my class, you will learn to fight and
defend yourself, but it will take years. And we will beat you up every night."
Real self defense (I recommend the books, especially the early books of Marc "Animal" MAcYoung of nononsenseselfdefense.com if this is your goal, and you can take his NSFW writing style) is not really about physical fighting skill. Its about awareness, avoidance, and so on. So if the goal is self defense, you have to ask yourself seriously if martial arts are the way to get there.
All of this is to say, judging the technical proficiency of a martial arts instructor is either impossible, moot, or won't get you very far.
This is why all serious answers about choosing an instructor or dojo or dojang or whatever it is called talk about how the instructor teaches their students, the culture of the dojo, and so on. That is what you will live with year in and year out.
If martial arts interest you, pick an instructor for these reasons, and begin practicing. If they are not good, you will very quickly find your learning at an end (a year or two) and can try another. If they are good, you will be on the road, and will start to develop the eye to let you see, and can eventually evaluate other instructors.