As a "safe" and "prudent" man, the narrator positions himself to be a both a casual and unbiased observer of Bartleby. Indeed, in the first few paragraphs, his tone and diction indicate that he is simply relating a story of curiosity, almost in the way a scientist might report on some observance of animal behavior. By clinging so tightly to this view of himself, the narrator is letting himself become more susceptible to Bartleby's influence. Other adults around him - Turkey and Nippers specifically - are tied so closely to their instincts and emotions that they can dismiss Bartelby, taking little interest in him in comparison to the other aspects of life that interest them more. By being "safe", the narrator is admitting that he has few or no emotional ties to his own life experience. This allows him to handle his job very effectively, very rationally. It also allows him to examine Bartleby very closely.
It is this unbiased examination, however, that brings change to the narrator. By closely observing a man of little emotion, a man lacking any desire or any strong attachment to the world around him, the narrator realizes he is closely observing himself. He's looking in a mirror. The narrator may be amused by the fluctuations in character and temper that Turkey and Nippers experience as a result of the indulgences, but he realizes that they at least have a tie to this world. They have heart and passion - for food and drink, but it is still a reason for existing and getting through the day. The narrator, like Bartleby, has only work. What would happen if he [the narrator] began to lose interest in work the way that Bartleby has? What would he do with his time then?