J. Donaldson, Reflections on the Harmony of Sensibility and Reason (1780), says the following:

An original delicacy of taste is also the inseparable effect and symptom of the true sensibility; which includes not only a sense of love, pity, gratitude, or common duty, for of those even the rudest natures are seldom altogether destitute; but it is a certain elegance of soul, which renders kindness most kind, and pleasure most pleasing; it is a genius and taste, the tenderness of friendship, the politeness of esteem, and the exquisite and refined endearments of love!
...
All ignorance of beauty, or depravity of taste, is defective animation; all improvement and perfection of these, is increased sensibility; the powers of the mind, as well as of the body, being rendered more perfect by a proper exercise of them.

Donaldson, at least, seems to think that "natural" taste is inborn, but can be taught to be better, and that improving someone's taste, gives them a correspondingly greater sensibility.