The narrator in Melville's tale does interact fully not only with Bartleby himself but with the other characters. He attempts to reason with Bartleby to discover the reason for the scrivener's unorthodox and increasingly alienated behavior. In fact, he seems to do this much more than would most others in his position, who would probably have fired Bartleby after the first or second reply of "I would prefer not to."
The narrator's interactions with people are of course not limited to Bartleby himself. At every juncture the lawyer intervenes with people—the other clerks in the office, the next occupier of the office, and the prison authorities—in order to protect Bartleby and save him from physical injury or death. The problem, if we can posit that Bartleby could have somehow been saved, is that the lawyer's actions are meant well, but are unhelpful.
From a twenty-first century perspective, it would be appropriate for the narrator to bring a physician to the office to examine Bartleby and offer some explanation for his behavior. Psychiatry as we know it did not yet exist at the time of the story (the 1850s), but physicians dealt in their way with mental disorders. However, the conditions of asylums at that time were appalling, with the patients often chained to the walls. So one cannot blame the lawyer for not reporting Bartleby's condition to the medical authorities.
One would almost think that a greater display of anger by the narrator might have yielded results, but this is to miss the point of the story. Bartleby has reached a point of alienation from which he cannot be brought back. In some sense the narrator understands this—that the hapless clerk is a symbol of despairing humanity, for whom no amount of interaction with others will do anything to relieve his suffering.