A nihilist is a person who believes that life has no meaning. A nihilist is a person who also, as such, rejects all moral codes as arbitrary.
When the older waiter is left alone in the café, he begins to contemplate the importance of a "clean and pleasant" place for people to go to when they are alone. He says that it is important to be able to sit down in a café, late at night, because otherwise the only option is to go to a bar or a bodega, and it is not possible to "stand before a bar with dignity." The older waiter then asks himself what it is that he fears about a life without dignity, and he says to himself that the thing he is afraid of is "a nothing he [knows] all too well." He says that life is "all nothing and a man [is] a nothing too." He says that life is "all nada," meaning that life has no intrinsic value or meaning. The implication seems to be that the older waiter thinks that, given that life has no meaning, it is reasonable to at least want to see one's existence out in a "clean and pleasant" and well-lighted place and with a little dignity.
Although the older waiter appears to be a thoroughgoing nihilist in that he professes to believe that life has no inherent meaning, he does, however, seem to place some meaning in and attach some value to the dignity of a life. He sympathizes with the old man who sits alone in the café at the beginning of the story, and he thinks it cruel of his fellow, younger waiter to send the old man home. He asks the younger waiter, "Why didn't you let him stay and drink?" He also says that he is always "reluctant to close up because there may be some one who needs the café." These sentiments betray an empathy and a sympathy that the old man has for others and particularly, it seems, for people who are lonely, like the old man who sits alone in the café. It is not possible to empathize with others if one attaches no value to or sees no meaning in the lives of those people. Therefore, while the older waiter may ostensibly seem to be a nihilist, he does in fact see significant value in the dignity of life.