from vocabulary.com:
"I myself am peculiarly susceptible to draughts."
Dr. Chasuble means that he, like Ernest who supposedly died of a severe chill, is susceptible to a current of air ("draught" is the British spelling of "draft"😉. Wilde is punning on "draught" to suggest Dr. Chasuble is also susceptible to "a serving of drink, usually alcoholic, drawn from a keg." This is evident by the previous line that "none of us are perfect." Another pun, although rather meaningless, could be on the British recognition of "draughts" as the game of checkers.