If you have a public persona, what do you think the chances are people don't already know you have a drinking problem and talk about it behind your back?
Those with the kind of heart to use it against you will find anything to try to make you look bad. As others have said, would you rather be known as a closet drunk or someone who's brave enough to tackle their problems?
I also have a high-profile, public facing career and that's part of what got me in so much trouble with alcohol, in the first place. The pressure, loneliness, and stress of trying to keep up an image of being perfect on the outside nearly destroyed me on the inside.
It took hitting rock bottom and trying to drag myself out that's been the most liberating, healthy, and best decision I've ever made. Bonus is the shame and fear people would "find out" what I was the only one refusing to see now puts me in a position to try to help someone else not experience the kind of stigma, embarrassment, and dread of asking for the help they (we) all may find ourselves at some point in life needing.
You could save someone's life simply by being honest about yours.
And, I've not attended AA. SR, CBT therapy, exercise, and eliminating anything or anyone toxic have been my tools for phase 2 of life's epic journey.