Green’s Dictionary of Slang

(orig. UK society) infatuated with.

1895		J.S. Wood Yale Yarns 155: He is dead crushed on that Louise Palfrey.
1896		S. Crane in N.Y. Journal 1 Nov. in Stallman (1966) 169: He’s crushed on Dollie Bangle.
a.1909		press cutting in J. Ware Passing Eng. of the Victorian Era 100/1: Quite new is the slang ‘crushed’. It is used in place of the expression, ‘mashed’, ‘struck’, etc., and is quite au fait with the summer resort girls. One hears everywhere murmurs of Charlie Binks being utterly ‘crushed’ on Mabel Banks, and so on with regard to various things. Dora tells Flora that she is ‘crushed’ on Jim’s new sailor, when she really isn’t damaging his headgear at all, and so it goes.
1929		J. Lait Broadway Melody 50: It was Babe who was crushed on Miss Starr, and who had enlisted the other two millionaires.