From the text, New York is the one who brought race into the interaction, not the bartender nor Jessie. I got the impression that the author sides with or at least wants to give the leeway to New York in this interaction, but I don't see why.

The author says that for New York, this might have been the most recent in a long string of race-based insults and slights, and "[t]hat neither Jessie nor I nor the bartender himself was in a position to say whether this had indeed been one of them, or why it felt like it to New York, or whether she was wrong."

So how is the bartender supposed to act in this scenario? He says, "No, sorry, she refused. I cannot. It's the law." and New York says, "Oh, a brown woman can't buy a drink?", and then he responds how? Castigating the bartender for doing his job makes him a scapegoat. If he'd said, "I don't serve black women cuz they get drunk too fast." or "I don't want your money." or something, I'd understand. But as it's written, this interaction tripped one of New York's nerves and she took it out on him and Jessie when they'd done nothing wrong.

Maybe that's the point: They'd done nothing wrong, but their very white-ness means it's still on them. There's no actionable course, then. Can't serve her because it's illegal. Can't not serve her because she's now in the mindset that it's because of her race.