The descriptions of these characters is also really part of setting. The office is cramped and uninspiring, and those who work there (with the possible exception of the lawyer, whose word we have to take for everything we read) could be considered misfits. Turkey and Nippers together make one effective worker, and Ginger Nuts is nothing but a glorified errand boy, though he's supposed to be an apprentice. Once we meet them, the tone of the office is set and helps prepare us for the inimitable Bartleby the scrivener who "would prefer not to."
In "Bartleby the Scrivener, A Tale of Wall Street," Melville chooses his order of character introduction, introducing the narrator's qualities, then the clerks' descriptions, and finally Bartleby, in order to illustrate what kind of man the lawyer is. The entire story depends upon the lawyer's reactions and responses to Bartleby and upon the reader accepting the lawyer's reactions, responses, and actions as wholly sincere and in keeping with his character.
Before introducing the conflict of the story, in the person of Bartleby, Melville establishes firmly that the lawyer is efficient, successful, respected among his professional peers and among a renowned clientele. Melville also establishes firmly that the lawyer is a man who not only wants a contention free life for himself but is willing to permit others the same privilege of a similarly contention free life, thus he observes and adjusts to his clerks peculiarities but never comments or tries to regulate said peculiarities. This is critical once Bartleby and his extraordinary peculiarities are introduced into the story.
I think that it is to show the everyday life of the narrator before Bartleby in introduced and to show that the narrator can and does deal with people's quirks as long as he understands them (he understands why turkey, nippers and ginger act the way they do.)
The narrator is already seen as someone who can deal with people's issues as long as they are explained while he can't deal with Bartleby because his actions are unexplainable.