The point referenced in the other answer to this question is a very good one, and it certainly pinpoints a climax in the story. However, there is another potential climax in the story or, at the very least, another moment at least as important as Mr. Doran's interview with Mrs. Mooney. This moment occurs at the very end of the tale, in which Polly is waiting for the results of her mother's interview with Mr. Doran. As she waits, Polly fantasizes about the future, and seems to think very positively about what might happen in a marriage to Mr. Doran. However, there is a change in tone in the final few lines of the story:
At last she heard her mother calling. She started to her feet and ran to the banisters.
"Come down, dear. Mr. Doran wants to speak to you."
Then she remembered what she had been waiting for. (75)
The abruptness of this passage's final line suggests that what Polly has been waiting for (her marriage to Mr. Doran) will be at odds with her apparently optimistic hopes for the future. In this moment, reality comes crashing down: this is not a lasting romance, but rather an arranged relationship designed to conform to social propriety, rather than honor true love. Thus, this becomes a moment of high tension and importance, as we get a glimpse into the reality of Mr. Doran and Polly's relationship and realize that, though both characters entertain hopes for the future, they are both trapped in an undesirable relationship from which there is no escape.
"The Boarding House" is yet another story by Joyce that is told through flashbacks, something Joyce did quite often. Because of this, it is difficult to decipher the question of a climax in many of his stories. However, if I had to pick a moment of highest tension, it would be near the end of the story when Mrs. Mooney has brought Doran in to discuss his affair with her daughter and to pressure him into marrying her. Previously, Doran has feebly assured Mrs. Mooney's daughter, Polly, that everything will be ok and that he'll do the right thing. He realizes, though, that he's been "had" because he's fallen victim to a seduction.