Your point is a possibility. According to the short story, they do not have children. This might suggest that they are young and for this reason may be poor. However, this is probably not the point that O. Henry is underlining.
I come to conclusion, because they each have one thing of value. To be more accurate, only James has something of worth - a watch. Della only has her hair. The fact that James only has one pocket watch of worth says that he comes from a poor family.
With this said, we should also point out that O. Henry is famous for writing about poor people, or working class people, who are doing through hard times.