How is Rip Van Winkle and his wife's relationship a "battle of the sexes"?
There is definitely a quality by which Rip Van Winkle's married life represents a domestic struggle. As a character, Rip is defined largely in terms of his laziness (much to the annoyance of his wife) and Irving refers to him as "a henpecked husband," which itself speaks volumes as to the nature of their relationship.
While Rip is described as good-natured but lazy, his wife is described as being domineering (and Irving even refers to her as in terms of being a "termagant" and a "terrible virago").
As we can see, just in the way the two are described, their personalities clash severely, and each tends to fit into gendered stereotypes. At the same time, however, if you're going to frame this marriage as a "battle of the sexes," it's also worth noting just how non-confrontational Rip himself is: he's not one to embrace conflict; quite on the contrary, he tries to flee it whenever he can.
Well, i think that one clue in answering this question is the response Rip Van Winkle has when he finds out his wife has died he feels the news "a drop of comfort." Clearly their relationship wasn't a positive one. You could say that ther relationship fits into the "battle of the sexes" in the typesd of things in which they argued about. Dame Van Winkle doesn't think Rip works hard enough, doesn't help around the house, and shows no interest in his children or wife. These are all common arguments and complaints. Rip, on the other hand, thinks his wife is being a "shrew." It seems to me that other woman would find Dame Van Winkle resonable, where other men might relate more to Rip.