How is it that Winterbourne has money but no class?

I do not think it can be said that Winterbourne of Henry James's Daisy Miller has "money but no class." On the contrary, he is the product of a particular kind of American family that has enough money to allow him to be "studying" in Geneva at the age of twenty-seven. Winterbourne has been educated in Geneva—both in terms of his school and his subsequent college education; there is, perhaps, some quiet element of disapproval from the narrative about the idea that he spends much of his time in Geneva pursuing foreign women, but there can be no doubt that Winterbourne is a upper-class American of good breeding.
In fact, one of the primary concerns of this novel is the extent to which Daisy, not Winterbourne, is perceived by the latter as having money but no class. Daisy's brother, Randolph, speaks in an uncultured way—he uses contractions such as "ain't"—and Winterbourne observes that Daisy has no embarrassment, as a better-bred lady might, at being spoken to by an unfamiliar young man. He thinks her face has "a want of finish" and, though she reins herself in to some extent, the lack of class in her family is betrayed by Randolph. Randolph's declaration, especially, that "my father's rich, you bet!" exemplifies how it can be that this family, while rich, is not of the same class as Winterbourne. Speaking about money was widely considered to be extremely gauche and crass in an era in which inherited money was equated with class, and new money, money made through business and goods, meant "nouveau riche," or classless wealth. It is to this latter circle that the Millers belong.