Is the teacher qualified? Anyone can claim to be a teacher, but may only know the basics, it's sad but true, you usually get what you pay for!
Does the teacher have a track record? Has he/she worked in a school perhaps, youth clubs, taught for the council or another authority? This will give you piece of mind that they are usually good if they have a track record of working for these places.
Has the teacher been criminal record checked (enhanced disclosure in the UK)? This isn't the easiest thing for a teacher to get, any real teacher should have one and be able to show you. Don't be embarrassed to ask!
Does the teacher have playing experience? Well, you wouldn't take tennis lessons of someone who has never played a match, would you?
These are just a few of the things. And remember, a great musician doesn't mean a great teacher!