This is a particularly astute question to ask about this short story, because I actually believe we cannot use traditional terms to analyse the characters. In most tales, there is a clearly defined central character--the protagonist--who is in conflict in some way with the antagonist who tries to prevent the protagonist from achieving his goals or aims. However, if we examine this story, there is no real character who stands in the way of Jim and Della and their love. In a sense, the central protagonist is Della, and, as the first paragraph of the story makes clear, her antagonist is not another character, but poverty:
One dollar and eighty-seven cents. That was all. And sixty cents of it was in pennies. Pennies saved one and two at a time by bulldozing the grocer and the vegetable man and the butcher until one's cheeks burned with the silent imputation of parsimony that such close dealing implied. Three times Della counted it. One dollar and eighty- seven cents.
Note the repetition of the paltry amount that Della has managed, through considerably hard work, to scrimp and save. It is poverty that thus tries to stand in Della's way and prevent her from obtaining the present she wants to get her husband as being a gift worthy of the love that she has him. Therefore, we can say that there is a protagonist and an antagonist, but only if we think more widely about these terms and are willing to classify poverty itself as the antagonist.