Character Role Analysis
None of the real characters is a candidate for antagonist. Plus, if you think about what it is that Della and Jim (the protagonists) struggle against, the only answer is poverty. Della doesn't have enough money to buy Jim the kind of present she wants to get for him, and that's what sets off all the problems in the story. Still, it's problematic to call poverty the antagonist, because it's not clear it's a bad a thing. You could say that it's Della and Jim's poverty which helps them focus on what matters – their love – and which forces them to make the sacrifice that proves their love.
If there is a real antagonist in this story, then, it might be better to say that it's money, or rather any worldview which makes money the most important thing. In this story, money represents the wrong way to value things. In one sense, the entire story is meant to make this point. It's not how much money something is worth that shows its real value as an object or as a gift. As we said, it's the lack of money that enables Della and Jim to understand the real meaning of giving. Money causes them to worry unnecessarily, and makes their life harder than it need be, when in fact they already have what matters.