The adjective—and its related words aesthete, aesthetical, etc.—begins with ae-. The variant esthetic is nonstandard and uncommon in both American and British English.
The word comes from the German Ästhetisch, popularized in translations of Kant, who used it in its original, scientific sense (the study of the perception of the senses), though the modern sense (concerning beauty, good taste, or the philosophy of aesthetics) was by then already more common.