There is a respect issue that appears to exist between the social classes in The Great Gatsby. Servants and lay people are treated as if they have no feelings. Wilson's wife is taken by a person of the upper classes (Tom) without any apparent remorse or moral twinge on Tom's part. Myrtle treats servants with a special derision as well.
This is a class-based problem and so can be called a social issue.
Add unfaithfulness to the list of social problems Fitzgerald addresses in The Great Gatsby. Rich men from the more fashionable and elite Egg come to his parties to meet women other than their wives. They are singers and movie starlets and generally "inferior" women from lower social and economic classes. Tom is not faithful, nor is Daisy--though Gatsby is willing to overlook Daisy's infidelity to Tom since she is with him. Myrtle is not faithful either, but it is somehow more forgivable for her, as she has a reason. She is trying to move up and improve her social and economic position, whereas the rich men simply seem unconcerned about their wives or their wedding vows.
The general lack of morality displayed in the novel is yet another social problem -- especially if you consider the more public morality issues. Gatsby is connected with the mob and is likely a bootlegger. He associates with a gangster like Wolfsheim who is rumored to have played a part in a gambling scandal with the World Series. Jordan is suspected of cheating in a golf tournament. Daisy leaves Myrtle for dead in the car accident and gets away with it. The corruption of this society is pervasive.
Another social problem that exists in Gatsby is the disillusionment of the young men especially after World War I. The way in which Nick Carraway regards people is indicative of this disillusionment. For, he often wonders why people say certain things and why others do not try to do things.
Another social problem present in this book is the problem of elitism and arrogance on the part of the rich. We see this in the way Daisy treats Gatsby back in Louisville. We see it in the way that the rich people drive so carelessly. We see it in the way that Tom uses Myrtle. I believe that Fitzgerald is criticising this sort of attitude on the part of the rich and identifying it as a major social problem of his time.
Part of what Fitzgerald sees as the problem in The Great Gatsby is that there was little, if any, attention paid to social, historical, or political issues in the time. The lifestyle of the characters is driven only by the selfish wants and desires of individuals. The historical context of prohibition might bring out one problem of the rise of the gangster lifestyle and the making of money through dubious means, something that we see in Gatsby himself. Additionally, I think that the fact that there is little attention paid to political reality might be a problem that Fitzgerald raises about the time period, itself. Since there seems to be so little attention paid to issues of solidarity and social cohesion, the argument is that this social order was doomed to failure. Fitzgerald might have been accurate, in that the excesses and indulgences of the Jazz Age played a large role in bringing on the Great Depression of the 1930s. This might be seen as another problem that is featured in the novel precisely because there is a lack of political or social concern amongst the characters in it.