Hilda is Judy Jones' nurse (a.k.a. her nanny) in the first section of "Winter Dreams." Hilda brings Judy to the Sherry Island Golf Club for golf lessons, apparently on Mortimer Jones' orders. Once they arrive, poor Hilda isn't sure where to go next.
This delay causes Judy Jones to throw a whopper of a tantrum. She wants to golf now! She wants a caddy now! When Hilda can't immediately produce either a golfing instructor or a caddy out of thin air, Judy tries to hit her in the chest with a golf club. Luckily, the caddy master finally arrives to take care of things. But what really matters in this scene is its unforgettable effect on the young Dexter.
Hilda is an employee of the Jones family. Even though, as Judy's nanny, she technically has some authority over Judy, Judy seems to feel absolute freedom to treat Hilda however she wants. Hilda is a servant, after all, and Judy has no qualms about treating her poorly. Judy shouts, "You damn little mean old thing!" (1.35) at Hilda even though, really, this situation is in no way Hilda's fault.
We can't help but be a bit disgusted by Judy's terrible treatment of Hilda (and anyone else who is socially lower than she is). But Dexter has quite a different reaction. Desperately wanting Judy for his own, he takes Judy's treatment of Hilda as proof of what he should not become (Hilda). He quits his job because he cannot be Judy's caddy: it will put him in a lower position than her, and he doesn't want to be her servant; he wants to be her man.