Good morning,
I started taking Goju-ryu karate with my kids back in May, and lucked out enough to choose one of the best schools I could imagine for both them and me (
Amatos Goju-ryu).
As always, I did my research on what to ask and how to evaluate. Here's what I discovered that was important to me and my family:
1.) Patience of the instructors. We were complete novices, but the senseis took time and time and time again to show us the correct way to do the moves. When we improved and asked for more toughness, they willingly obliged.
Thus, I would ask the instructors, what's your mode of teaching? If I want to be treated tougher/softer, will that be honored? (the tougher part turned out to be very important to me, 3 months later.... I love pushing myself now that my confidence is high. And my kids....wow, one of my girls went up to the sensei a few months ago and said, you treat me like a girl, can you be more difficult with me? He was, and she was thrilled).
2.) Experience of the instructors. The senseis are 3rd degree and above - the two main beginner senseis are 5th degree. But experience by itself isn't as useful as experience + teaching ability.
I would ask, how long have your senseis been teaching? From whom did they learn?
3.) Machismo attitude (for lack of a better word). Watch a class or two - is there any 'bully' factor from any of the students? If so, does the sensei put a firm stop to it?
I would ask, how do your senseis stop bullying? Remember, that can occur in adult classes as well.
4.) Payments. Can you take a lesson or two for free to 'try it out'? Are contracts required? If so, is it cheaper to pay by year? Are uniforms included? What about belt promotions, do you pay for that as well? A useful resource I found regarding dojos and contracts is the forum
Martial Arts BS, Fraud, Investigations, and Standards.
5.) Levels training. Visit the dojo - do you see only lower and only higher ranked belts, but nobody in between? That would indicate people tend to drop out after the first few belts. If so, I'd wonder, why didn't they stay?
6.) Katas. Okay, this is a personal bias of mine.

Lots of karate/martial arts places put the emphasis on 'self defense' and 'karate'....and ignore the benefits of katas. Not everything is fighting; katas, while not blindingly fast, really can help give your muscles the muscle memory useful for more advanced techniques.
These are what I can think of off the top of me head - hope it helps!
Best wishes,
Barbara