Green’s Dictionary of Slang

(US) to develop a dislike for or grudge against, to be hostile or oppressive to.

1875		C.B. Lewis Quod’s Odds 381: The adult male population [...] got down on [him] [DA].
1898		E.N. Westcott David Harum 105: Dave got down on him fer some little thing or other, an’ he’s got his walkin’ papers.
1952		J. Jones From Here to Eternity (1998) 649: It did not matter that it was the queer investigation that had partly caused the NCO School to get down on him.
1965		C. Brown Manchild in the Promised Land (1969) 174: Mama used to get down on me about hanging out with Reno.
1970		R.D. Abrahams Positively Black 126: [I]t is one's friends who take one away from the home and who ‘badmouth’ one’s partner. In such situations a number of traditional arguments are given by friends: [. . .] ‘she’s getting down on you to her friends,’ ‘you can’t trust any of them,’ and so on.
1985		O. Hawkins Chili 36: It was kinda phoney from the git, me gettin’ down on her case so bad.