Dexter's characterization throughout the story largely depends on his thoughts and emotions, not on his physical appearance. But we know for certain that he dresses well, and we can infer that he's physically fit and reasonably handsome.
Throughout Parts I and II of the story, we get very little physical description of Dexter aside from the narrator's descriptions of his motions, like when he stands still in awe of Judy, or like when he stretches out on his springboard in his bathing suit after swimming. (We're practically deluged with physical descriptions of Judy, though!) The fact that Dexter is a very capable caddy as a teenager, then a capable golfer as a young adult, does suggest that he has some degree of physical fitness.
However, Part III provides this image of Dexter:
He knew the sort of men they were--the men who when he first went to college had entered from the great prep schools with graceful clothes and the deep tan of healthy summers. He had seen that, in one sense, he was better than these men. He was newer and stronger. Yet in acknowledging to himself that he wished his children to be like them he was admitting that he was but the rough, strong stuff from which they eternally sprang.
From the information above, you can infer that although Dexter isn't quite as good-looking or tan as Judy's other boyfriends, he's still fit and strong. We also find out that Dexter dresses very sharply:
When the time had come for him to wear good clothes, he had known who were the best tailors in America, and the best tailors in America had made him the suit he wore this evening.
Later, in Part IV of the story, Judy comments on Dexter's appearance:
"You're handsomer than you used to be," she said thoughtfully. "Dexter, you have the most rememberable eyes."
Although it makes sense that Judy would only associate with handsome men, meaning Dexter must be handsome, we also can't put much faith in her words. She often says what she doesn't mean, and she's very manipulative.