Which specific elements in Crane's "The Bride Comes to Yellow Sky" help to determine its theme, and how?

The central theme in "The Bride Goes to Yellow Sky" is the passing away of the Old West to make way for the future. The two elements which most emphasize this theme are the setting and characters.
The first scenes in the story are set on a train coming from San Antonio. It is a symbol of the fast-paced modern world. Jack Marshall and his bride are out of place on the train, uncomfortable in their wedding clothes and mocked by both the other passengers and the porters. Yellow Sky on the other hand is a frontier town, rougher in its manners. The train coming to Yellow Sky represents modernity catching up to the Old West and replacing it.
The characters are plays on Old West tropes: the law-abiding marshall bent on justice, the good woman with domestic influence, and the trigger-happy criminal sworn on vengeance. However, they are stripped of all their romance. Jack is more terrified of what the town will think of his marrying a woman without their consent than of any criminals that might threaten his new bride. His bride is a timid, middle-aged woman, not the youthful beauty one might expect.
Scratchy Wilson is the least glamorous of all. Initially presented as a menace, by the end of the story he is rendered pathetic. When he learns Jack is now married and has no time for their old cat-and-mouse game, he reacts like a little boy being told his best friend cannot play ball with him anymore. In fact, he is referred to as a child-like figure throughout, even though he asserts to Jack "I ain't no kid."
This last, anticlimactic defeat of Scratchy Wilson becomes a representation for how the wild ways of the Old West were being tamed by "civilization" in the late nineteenth century.

Crane's famous tale is a parable about the contrast and the clash between the old and new, the primitive and the civilized.
In the story, Potter is traveling by train with his bride from San Antonio to a remote town in Texas. The description of the interior of the railway carriage, and the mere fact of journeying by train, are meant to emphasize the new, civilized side of the frontier. The porters are overly attentive to Potter and his bride; the fixtures are of polished brass and wood that gleams "like oil;" the dinner costs a whole dollar. The disconnect between this new world of luxury and ease, and the wild frontier of Yellow Sky, is also evident in Potter's anxiety over how the town will receive his bride, about whom he has told them nothing. Potter himself isn't quite at ease with the advanced, comfortable world from which he's bringing her. He gives the porter a tip "as he had seen others do."
The theme of culture clash between old and new is further delineated in the rather comical description of the bar in Yellow Sky and the way everyone cowers before Scratchy Wilson. The town drunk who gets out of hand became a cliché of Western literature and film. If Scratchy represents the old untamed West, then his defeat, the vanishing of his species, so to speak, comes about when he sees that Potter has a bride with him. It's as if Potter's married status represents civilization. The title of Crane's story is emblematic of the taming of the Old West. When the bride comes to Yellow Sky, it's the beginning of the end for Scratchy and the wildness of the frontier, as Scratchy walks away, foregoing the planned fight with Potter.

Stephen Crane's 1898 short story was meant to explore, sometimes humorously, how the American West was changing near the turn of that century. 
Trains were making it easier for people to travel great distances, and Crane opens the story with the town policeman, Jack Potter, returning to Yellow Sky with the woman he has married earlier that day in San Antonio. She is an Easterner unused to the vast distances of the West and the unique pressures on her husband in his prominent role in Yellow Sky. So, the initial setting of the story, the train, illustrates how trains were bringing Easterners and Westerners together.
The fact that the story's protagonist is a lawman with enemies in a Western town is a familiar trope. Potter really has two "enemies:" the town, to whom he will have to answer for his choice in marrying an Easterner without their "permission;" and Scratchy, his "ancient enemy" and a living artifact of the Old West.
The fact that Scratchy walks away from his "showdown" with Jack Potter—who is described as a "city policeman," and not a sheriff—suggests that civilization and modern thinking have reached West Texas.