Will this criminal record cause the US Embassy to not approve my Filipino fiancé's K1 visa application?

You have not identified whether you or your fiance has a criminal record and you have not identified what that criminal record may be. There really is no substitute for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.