One way to identify the features of the respective social classes of Miss Churm, Oronte, and the Monarchs would be to think about how the features of the characters reflect their implied social class.
Henry James introduces the Monarchs first, so perhaps it’d be best to start with them. The painter who narrates the story depicts the couple as shy and reserved. They behave in such a secretive way that the painter thinks that the Monarchs aren’t really husband and wife.
The Monarchs’ detached style could be a symptom of a social class that prides itself on coming across as mannered, intellectual, and, in a way, cool. Like their social class, the Monarchs are ordered and tidy. At one point, the painter imagines the Monarchs’ life. He thinks about their tweeds, their smoking room, and their “neat umbrellas.”
Speaking of umbrellas, Miss Churm makes her appearance carrying a wet umbrella. The wet umbrella contrasts with the Monarchs’ “neat umbrellas.” This tiny detail could be significant. It might be James’s way of distinguishing between Miss Churm’s social class and that of the Monarchs. The reader can tell that Miss Churm comes from a different social class, because she is not so ordered, neat, shy, or reserved. She is a talkative beer drinker with a “whimsical sensibility.” Miss Churm’s inability to spell also puts her in a different class than the “smart” Mrs. Monarch.
As for Oronte, it’s reasonable to deduce that he’s from the same social class as Miss Churm since he’s as “good as Miss Churm.” In James’s story, at least in the context of art, it’s as if the lower classes have malleable features while the upper classes—or those with upper-class pretensions (remember, the Monarchs have lost most of their money)—have rigid and stiff features. The lower classes can be anything; the upper classes are constrained.