I don’t think their is any way you can explain to people that you are a responsible gambler, most of them won’t want to listen to you, and I don’t think you should start telling everybody your story that you are a responsible gambler, the best thing you can do is just to leave them Anything anybody says about you, just over look it and move one, nobody is a saint in this life. In my country if people notice that you are a gambler they believe you are no more responsible, which is very wrong, I don’t see anything wrong in gambling, the only thing I can say is bad is been addicted to gambling, but if you are not addicted then I don’t see anything wrong in gambling but people won’t want to listen.
The best thing that you can do to let them know and believe that you are a responsible gambler is by practicing it instead of saying it or rubbing it to their face. No one will really believe if you'll just say it. You should prove it to them by actions and not by words. Although it's not really your responsibility to prove yourself and on how responsible you are. But if you want to, through actions would be better because they can see the benefits of gambling and would not just focus on its repercussions.
There is really a stigma in gambling. Once people knew that you are a gambler, they will view you in another perspective such as being irresponsible, addict, or even a lazy fella. We can't really blame them because that's how the reputation of gambling in the mainstream media is. Since most people don't understand the need of other people to gamble, they just view them as lazy individuals and irresponsible ones because instead of focusing on what they deem important, they focus more in gambling. Some can't grasp the idea that we have different priorities which make them easily conclude such.