A lot of elements can go into this question, so I figured I'd start with the necessary conditions. It's a short list:
- Your instructor must know more than you. Not be better than you - someone might have sloppy technique because of physical infirmity, time off, whatever, but have the intellectual knowledge needed to help you advance. Now many people would want an instructor who is a master of their discipline, but that isn't necessary, especially for a beginning or intermediate student. In my old club we often had third or fourth kyu students leading the beginners through their techniques. In all honesty, a green belt could do as good a job showing someone who knew nothing how to make a fist or stand in sanchin as our chief instructor - sometimes better, maybe because they'd "been there" more recently. The more advanced students needed more advanced help, of course. But really, as long as your instructor is a few steps ahead of where you are they can help you advance.
- Your instructor must be able to assess your technique and break down what's wrong with it. Some people are naturally gifted - they see a kick, and they move their bodies into that position. They might not realize consciously that they're pivoting on the supporting foot. When these people see someone struggling they might not be able to understand what's going wrong. They see that the kick is "bad," know it should be done better, but can't tell yo what to fix in particular. That person may have very good karate but isn't a good instructor.
- Be able to give constructive feedback, or coaching cues, relative to the mistakes you're making. This can be verbal, but I've seen some instructors see a flaw in my technique and then just show it to me in such a way that I could see the specific problem. A language barrier isn't insurmountable, although the ability to give good verbal feedback and coaching cues is probably a good thing.
I originally stopped my list there, but I've had some second thoughts (and I might have some third thoughts later). I'll add one more to the list: your instructor must know enough about proper technique and safety to limit the injuries that occur in class. If your instructor forces you to push to the point where you are injured very often or permanently damaged or have to quit then you won't make progress.
Now, having a list of necessary conditions is all well and good: I think this represents a good start on the bare minimum qualifications for an instructor to have if you're going to make progress. I considered, but left off, a few things other people might include, such as issues of character. Could you learn karate from a jerk? I think you could. It might not be ideal, but it's not impossible. If you're an adult, I think it might not matter much at all. You might not enjoy the experience, but you could still develop very good skills.
Is this list also a set of sufficient conditions for a good instructor? I'd say probably not. An instructor with just these qualifications would be enough to guide your progress, but you'd want to see more from a good teacher. Maybe an inspirational demeanor. Good knowledge of how to structure a workout to maximize its benefits. A knowledge of the history of the art that he/she could pass on to you. Impressive physical skills, which can both inspire and demonstrate for you.
A lot of what we need from an instructor is also relative. I'm almost forty. I don't need a teacher with good management skills or patience. My two year old son, however, will need a very patient teacher with strong kid-skills when he starts training. A beginner doesn't need a teacher with advanced knowledge. At my age and rank most fourth kyu students aren't going to be able to spot my mistakes, or at least not all of them. Motivational skills are another issue. I'm pretty self-motivated. If my daughter were to train, she'd need a teacher who was really good at getting her to want to train, or she'd give up very quickly. I'd also want my kids to learn from somebody who was something of a role model in general - someone fit, with good ethics, and a calm demeanor. For myself, it's not as important. My teacher is an excellent person, but I don't feel that she needs to be to guide my personal moral development.
Please comment if you can think of other attributes to add. I consider this a work in progress.