From Mental Floss:
YOU BETTER WATCH OUT, YOU BETTER NOT CRY
That’s right, Santa Claus is coming to town, so you better watch out. Or is it “you’d better watch out?” Many grammar guides advise that the proper form is “you’d better” because the construction comes from “you had better,” and it doesn’t make sense without the “had.” The problem is, it doesn’t make much sense with the “had” either, if you want to do a picky word-by-word breakdown.
Though the “had better” construction has been a part of English for 1000 years, it came from a distortion of phrases like “him were better that he never were born,” where “were” was a subjunctive (“it would have been better”) and “him” (or “me,” “you,” “us”) was in the dative case (“him were better” = “it would have been better for him”). People started changing the dative to the subject case (“he were better”) and then changed the “were” to “had.”
That was all hundreds of years ago. Then, in the 1800s, people started dropping the “had.” The grammar books of the late 1800s tried mightily to shore up the “had” (some even making up a rule from nowhere that it should be “would,” as in “he would better”), but these days the bare form is considered correct, if a bit casual for formal contexts. Clearly, “Santa Claus is Coming to Town” wants nothing to do with fancy formality. So “you better watch out” is the way to go.