In my reading of history, people from prominent families in New York, Boston, and other areas of the country figure very prominently from the American revolution to about the 1920s. For example, the Lodge, Shaw, Adams, and Peabody families (to name a few).
After that point, it seems far more rare to see influential players who come from these families. Did they die out? Did wealth spread so thinly through the generations that they lost their influence? Did the Great Depression remove their significance?
The Boston Brahmins refer to a number of prominent Boston families, so there's no one answer as to "what happened" to them. Many of their descendants are still highly influential, especially in the world of politics (John Kerry--member of the Forbes family, Lincoln Chafee).
Others managed to continue to build the family wealth. The Cabot Corporation, founded in 1882, was valued at $1.5 billion in 1995, with 120 Cabot descendants owning 1/5 of the company stock (New York Times obit of Thomas Cabot).
This Boston Magazine article looks at what they call the "new Brahmins," wealthy Irish, Jewish, and Italian families that now fill that philanthropical and influential role. The article also lists a number of Brahmin descendants that are still active in Boston society.