Thematically, this is not the most important aspect of "Skunk Hour," but it's one that's clearly there, especially from the beginning, so it's worth our attention. Social status also seems to be one of many reasons our speaker feels so distant from the other islanders – they're caught up in money and status while he couldn't seem to care less. Check out the beginning of the poem. Before the speaker goes all inward and dark on us, he's painting a picture of the fancy, bourgeois lifestyle (or the desire for it), old money (and wanting to keep the newbies out), and the vacationers that use the island as their summer playground then duck out when the weather turns chilly. With that much attention to the issue of class at the beginning of the poem, it's certainly an important part of Nautilus Island's fabric.
Lowell is just poking fun at the old-fashioned preoccupation with class. The social status is all in the heads of these characters, and doesn't really exist in real life.
Class issues and envy are the reasons for our speaker's feelings of loneliness and madness.