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 to English 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), even though the modern sense (concerning beauty, good taste, or the philosophy of aesthetics) was already more common.