In the long run, the problem is not with rich people consuming too much. It is with too many people becoming rich. As global levels of affluence rise the greater number of affluent people will put more strain on nature than the excesses of the already rich.
Paradoxically, the problem is the solution. As people become more affluent they have fewer children. Well regulated capitalism is the key to lifting people out of poverty, which will in turn lower the birthrates, which will reduce the environmental burden.
To achieve environmental sustainability, the best bet is to help the global poor be self-sufficient and economically secure.