I looked into this several years ago, as I have teaching experience and always had many people asking for tutoring. Unfortunately, you can't just set yourself up as a private tutor. To do it legally, you would need to form a legitimate business with minimum required capital investment, Thai shareholders, Thai employees (at least 2 if you are legally married to a Thai citizen and 4 if not). You would also have to adhere to certain accounting practices, which adds another layer of costs. The company you set up would then have to apply for a work permit for you in order for you to be able to teach. Your business would also need approval from the Ministry of Education. They have all sorts of rules about the size of the premises you must have to teach out of, number of bathrooms, waiting area, etc. As you can see, things get complicated and costly. I know several foreigners with Thai spouses that opened up tutoring centers. All of them wound up getting out of it after a few years, saying that the income from it just wasn't worth the hassle. Of course, you'll see many tutoring centers that are run by Thai teachers that don't seem to meet all the requirements. I think that they can get around things because the rules are applied differently for them.
Many foreigners teach/tutor illegally. There are risks to that. I have actually heard immigration officers asking children in one neighborhood where I lived if any of their foreign neighbors were teaching them English. The IO's you see when you do your extensions are essentially office workers. There are other ones whose main job is to seek out and arrest foreigners who are working illegally. There's also a lot of jealousy here and it would be very easy for someone to report you. A foreigner always stands out here and is like low hanging fruit for the cops.
It's unfortunate that it's like this because there are many areas where the local folks don't have access to competent tutors. Or, maybe you just want to help some of the local kids out and make a little money on the side. That probably won't give you anywhere near enough income to cover all the operating costs, if done legally. If there are any tutoring centers where you live, you can check with them, as some will employ foreign teachers.