The middle name Dillingham plus the fact that Jim inherited his gold watch from his father who had inherited it from Jim's maternal grandfather suggest that Jim comes from a social class superior to that to which he and Della now belong. The name Dillingham in itself does not necessarily mean that Jim comes from a higher class, but the fact that O. Henry puts it in the story and emphasizes it shows the author intended to imply something--if only to explain how a man in Jim's position could come to own an expensive gold watch. Jim values his watch because it is a reminder of his ancestry. That is why he is always taking it out and looking at it. The fact that he is always pulling his watch out of his pocket suggests that there is a mathematical probability of his dropping it sooner or later. That was why men had watch fobs for pocket watches, and that it is why it occurred to Della to buy Jim a watch fob.
Della would seem to come from a working-class background. Here is a sample of her speech:
“Cut it off and sold it,” said Della. “Don't you like me just as well, anyhow? I'm me without my hair, ain't I?”
Della not only seems to love Jim deeply, but she also seems to look up to him as to a person of a higher social class than herself. She seems to feel privileged to be married to him. He would never use the word "ain't." She wants to buy him a present that would be "worthy" of him, which could mean worthy of a man from his social background.
Jim's character seems to reflect higher social antecedents. He is quiet, reserved, polite, considerate, soft-spoken. Della loves him for being a gentleman as well as for other reasons.