It depends what you mean. If the emphasis is on the sort of coffee (brand, whether it's fresh, powdered, or freeze-dried) then "Mi trinkas nenian kafon" would be fine. But if the emphasis id on the fact that you don't drink coffee (but drink tea, water or whatever instead), then how about "Mi neniam trinkas kafon" or "Mi tute ne trinkas kafon"? I suppose there's a third possible meaning: "I don't drink any amount of coffee." For that, "Mi trinkas nenion da kafo" is my suggestion.
I feel like this is probably an English thing - but it sounds strange to me to say a positive to a negative. "I drink... No kind of coffee". I'd rather phrase by saying "Mi ne trinkas ĉian kafon". I'm not sure if that's grammatically correct though.
"Mi ne trinkas ĉian kafon" means "I drink some kinds of coffee but not all kinds".
Correlatives starting with "neni" negate the verb. No need to add "ne".