The officer was the "childhood sweetheart" (1.4) of the narrator's wife and her first husband. The narrator is jealous of him, and perhaps a little angry with him, too. We sense some hostility in these lines:
Her officer—why should he have a name? he was the childhood sweetheart, and what more does he want? came home from somewhere, found her, and called the ambulance. (1.5)
Of course, he's describing his wife's suicide attempt. Like Beulah, the officer helps expose other characters. He helps us see the narrator's jealousy, his interest in his wife's past, and possibly that he blames the officer for his wife's past misery. The officer gives us insight into the narrator's wife as well, showing her as a sensitive, troubled person. Like Beulah, he's a foil for the narrator. Life with the narrator is supposed to be a contrast to his wife's life with the officer. But, the narrator is in danger of becoming ex-husband number two if something doesn't change soon.