Once you have a little background, this answer is really a surprisingly simple one. Let’s start with the definition of farther (f-a-r-t-h-e-r). American Heritage Dictionary says farther is the comparative form of far: far, farther, farthest. Farther is a generalized, nonspecific measure of distance.
If someone has their side view turned to you, i.e., is not looking directly at you, one of their hands is closer to you and the other farther from you. Let’s say the person is standing before you in profile and looking out an open window to your right. This would mean that their right hand is closer to you. It also would mean that their left is farther from you. This same explanation applies if they are in a three-quarter turn toward you; three-quarters is a position (with variations, of course) half way in between profile and full view. In a three-quarters view, still looking out the open window to your right, the closer hand is still the right and the farther hand is still the left.
If the person turns around and now gazes at the vase of peonies to your left, the closer hand is now the left, while the farther hand is now the right. In the context of Wilbur’s “Playboy” (an ironically satirical title), the dunce of a boy sitting on the stool is entranced with a photograph in a magazine. The woman in the photograph is posed in a three-quarters turn. Thus she has a closer hand and a farther hand. Whichever direction her pose positions her, whether looking toward dunce boy’s left, as in our vase of peonies example, or looking toward his right, as in our open window example, the woman’s hand that is holding the goblet is the one on the side farther away from the dunce boy and is thus the farther hand.
Amidst which, kneeling in a supple pose,
She lifts a goblet in her farther hand,
As if about to toast a flower-stand
Above which hovers an exploding rose.