Henry James wrote Daisy Miller, according to his biography, after hearing how some European socialites spoke with contempt against the mannerisms, lack of culture, and lack of social status of a nouveau riche socialite who was trying to rub shoulders with rich aristocrats during her first grand tour of Europe. At the time of James, for an American family to come to money was a demonstration of how differently Americans and Europeans view the making of a person: In Europe, you need peerage and name. In America, money is enough to make you socially acceptable.
In the case of James, he used this story he heard to convey a message: How society views a person who has just come from somewhere else using the assumptions of their current society, and how individuals observe that person as well.
Therefore, the "study" in two parts is not only on the views of how society shun Daisy Miller for her nouveau riche behavior, but also how Winterbourne saw her under his criteria of a man of society, and as a man who could possibly end up loving a controversial woman like that.
The incident that triggered "Daisy Miller" was a story James heard about a young American socialite, travelling in Italy, who fell for an Italian of uncertain social standing. So the impetus behind James' story was, in part, about class and social customs. Americans, coming from a society where social class is tied directly to wealth, often did not understand how, in Europe, social standing depended more on heredity. In fact, from the point of view of upper class Europeans, people like the socialite who inspired James were at best clueless about social forms and, at worst, simply vulgar.
So, one reason James wrote the story was to dramatize the conflict American and Eurpoean attitudes towards class could create. But, if that were all James had in mind, I don't think his story would have endured as it has.
To me, the real reason behind the story is to try to show the nature of Winterbourne's attraction to Daisy and what it is that keeps them apart. Part of this has to do with Daisy's essential "otherness"—she is, for Winterbourne and for the reader, a delectable mystery. It is because her internal life is such a blank that Winterbourne and the reader are able to project all sorts of motivations and attitudes on her. In this sense, the story is not really about Daisy at all, but about Winterbourne and his complex reaction to her, which shows him, finally, to take sides against Daisy and her "impropriety."