The industry may have created that phrase in a conveniently ambiguous way; still, in English, positives are always better than negatives when giving a command. Consider the difference between "I don't eat junk food" and "I always eat healthy foods." The first contains the proposition that you have the capability of eating junk foods; the second does not.

Perhaps "Always Drive Safely" or "Always be Responsible" as the link text would be better?

I completely agree that the message "Always drink responsibly" is focused on escaping legal problems while getting customers to drink more. That said, this site has exactly the same goals as the alcohol industry - to get their clients to drink more, and use this product in conjunction with that drinking. So, perhaps "Always drink responsibly" is actually what they want.

Good observation.