Inspired to write "The Ones Who Walk Away from Omelas" partly as a response to utilitarianism, Ursula K. Le Guin creates a fantasy land called Omelas where smiles have become "archaic." 

In Omelas, smiles have become antiquated because happiness appears to be universal now. However, this appearance of a utopia is false because there is a condition attached to the residents' happiness. Hidden away is a miserable child who bears the burden of immorality and selfishness for all the other residents of Omelas. When certain residents of Omelas learn of the existence of this child, they leave Omelas. Others are more pragmatic, as they realize that the "nobility of their architecture, the poignancy of their music, the profundity of their science" would not exist without the child's being in its present degraded position. They believe in "the greater good."

This belief that an individual or some individuals must suffer so that others can prosper and be happy was a part of the ideology that acted as a driving force for Nazism. In Germany, Jews, especially the wealthy and prosperous, were targeted as the cause of the economic failure of the country. If the "Jewish Question" [Die Judenfrage] were dealt with, then Germany could prosper, the Nazis contended. History, of course, teaches that such an idea was incredibly inhumane.

In a democracy, each individual is of importance. Therefore, any such coldly "pragmatic" solution as that of Omelas's incarcerating one being to bear the sins of all is indefensible, because the acceptance of the scapegoated child's miserable state is morally revolting. Walking away from the problem is also an unsatisfactory solution. In a democracy, the only socially sustainable solution is for all people to be treated equitably. So, all people must endure some misfortune and unhappiness in their lives, and no one should be victimized. Ironically, it can be argued that the only way to know any true happiness is to actually have suffered some in one's lifetime. Certainly, then, if people suffer and survive misfortune, smiles need not be "archaic."

A socially-just solution to the problem in "The Ones Who Walk Away from Omelas" would address the suffering of the child.

A socially-just solution represents a desire for equity.  It strives to provide fairness for all people in a social setting. This would mean that the inherent unfairness in Omelas would have to be addressed. For example, freeing the child from being imprisoned would be one socially-just solution.

Le Guin's story is predicated upon the idea that the people in Omelas can only be happy because the child suffers.  A solution predicated upon social justice might involve freeing the child, would remedy this predicament.   It would be rooted in fairness for all people and would seek to reverse institutional unfairness.  The child would experience the same freedom that the people of Omelas experience. This mindset is socially-just because it challenges how the townspeople see the child. The townspeople know that they can only be happy because the child is miserable.  A socially-just solution would require be one that is fair to all people.    

In seeking to eliminate the child's suffering, a socially-just solution would be concerned with equity for all.  Regardless of the the happiness of other people, it would emphatically suggest that the child's condition has to be remedied. When this is done, the society of Omelas is closer to being more just because it moves closer to fairness.