As far as ESL job requirements to legally work in Thailand go, the most non-negotiable item is a 4-year college degree. In nearly all cases, at the very least, you will be asked to produce photocopies of your degree, which in some cases may be enough to satisfy the Thai bureaucrat processing your work permit. In others, you might be asked for the original diploma.
The type of degree, except in higher education environments, is irrelevant. Indeed, many schools at the high school level and below are not themselves concerned with the degree requirement and view it more as an annoying impediment to hiring foreign teachers than as a legitimate requirement for an ESL job in their school.
Most institutions process all the necessary documents for the work visa on the foreign teacher’s behalf, including necessary translations to Thai. If you want to do it ahead of time, though, you can use an online translation service like Tomedes.