I think that this can be taken in several ways. On one hand, there is a definite notion of social and economic advancement in the new world that Williams depicts. In this realm, the immigrants from other world such as the Kowalskis could come to America without landed wealth or privilege and "make it." This vision of the American Dream is one where there is no hierarchy or notion of elitism present. Part of Stanley's intense dislike of Blanche is because he sees her as "old world," and representing the forces of social stratification that would keep a person like Stanley at the lowest rung of society. When he makes comments such as how he was able to pull Stella down from "those high columns" that adorned Blanche's old home, there is a deliberate disdain for that older social setting. At the same time, it is really important to never take anything that Williams writes on face value. Nothing is simple in his work. While there is a pro- democratic setting present, Williams might also be suggesting that the unlimited freedom and liberty present can help to justify a lack of moral structure or order within the pursuit of the American Dream. Stanley might be an embodiment of the American Dream of success, but his cruelty to Blanche and his mannerism of a brute might raise question to whether this dream is worthy of pursuit. Typical of Williams, thought, the reader is trapped between two predicaments, both of which are extremely painful and difficult, with no relief in sight.
I agree with Akannan's answer( it was an excellent answer). I would just like to add that for a women the American dream is harder to pursue since women didn't have the same social and professional possibilities as men. Man are not judged if they behave like Stanley but women are.