Hawthorne himself worked as a customs officer, so there's more than a hint of autobiography here. The narrator is chief executive of the Custom House, a place where taxes are paid on imported goods. It's clear from the unflattering description he gives of his colleagues—"wearisome old souls"—that he doesn't have much time for them. This would appear to indicate that in the intervening centuries since Hester Prynne passed away, nothing much has changed about Salem and its people.
This impression is further reinforced by the narrator's description of the town being old and run-down. It's notable, too, that the scarlet letter itself— the very same one that Hester was forced to wear all those years ago—still survives, wrapped up in an old package that's been gathering dust at the Custom House. The Puritans who persecuted Hester are long gone, but the symbol of her persecution still lives on in a Custom House staffed by "wearisome old souls" in a decaying, declining town.