Connecticut must suck. I don't know the exact clinical diagnosis of alcoholism, but the best diagnosis I know is that if one takes one drink, one cannot predict with certainty what will happen after that drink. If you've been to a treatment program, then you are probably an alcoholic. Once an alcoholic, always an alcoholic. Therefore, I don't know what Connecticut means when it says "current". Since you've been diagnosed as one, then you are currently (and always will be) an alcoholic. You don't become "cured". Connecticut may mean that if you are currently drinking; if you're not doing so now (most places say that if you've not been drinking for a certain amount of time, and are currently participating in some sort of continuing program (such as AA), then you're not an active alcoholic, and should be eligible for a license). So, Connecticut needs to be more specific in its laws and definition. Do you feel like taking them to court, and challenging the law? That'll take time and money for a lawyer (if you can find one who is willing to take on the state, and possibly appeals to a higher court, etc.) If you find one, you might try stating that you don't have a current diagnosis. If they find out you've been diagnosed as such, and prosecute you or take your license away, then you might have a case worth fighting for. (Of course, I'm not a lawyer; got any frinds who are? Then you might ask them.)
In summary, I don't know what to do. I think I'd say "no", and take my chances.