On the whole, America has become a much less racially stratified society than it was in the 1940s, '50s, or '60s. You can see indications of this all over the place, starting with the fact that we have a black man working in the Oval Office. At the same time, pure income inequality has become much bigger and on a variety of different fronts income-linked stratification has become a bigger deal. One way in which this reflects itself is that the "achievement gap" in school between white kids and black kids used to be bigger than the gap between rich and poor. An excellent Sabrina Tavernise article (featuring the chart I've poached) touting research from Stanford's Sean Reardon makes the point very clearly.
Race and class in the United States are obviously linked phenomena, but also to some extent separate ones. They've both mattered throughout American history, but over time race has tended to be very important and in the middle of the 20th century class was at something of a historic low point in importance. But over the past generation the basic forecasts of William Julius Wilson's The Declining Significance of Race have tended to be vindicated as you see here.
Reardon's abstract is extremely interesting and I'm looking forward to reading the whole paper. Here are some points you won't find in the chart and that cut a bit against the grain:
First, the income achievement gap (defined here as the income difference between a child from a family at the 90th percentile of the family income distribution and a child from a family at the 10th percentile) is now nearly twice as large as the black-white achievement gap. Fifty years ago, in contrast, the black-white gap was one and a half to two times as large as the income gap. Second, as Greg Duncan and Katherine Magnuson note in chapter 3 of this volume, the income achievement gap is large when children enter kindergarten and does not appear to grow (or narrow) appreciably as children progress through school. Third, although rising income inequality may play a role in the growing income achievement gap, it does not appear to be the dominant factor. The gap appears to have grown at least partly because of an increase in the association between family income and children’s academic achievement for families above the median income level: a given difference in family incomes now corresponds to a 30 to 60 percent larger difference in achievement than it did for children born in the 1970s. Moreover, evidence from other studies suggests that this may be in part a result of increasing parental investment in children’s cognitive development. Finally, the growing income achievement gap does not appear to be a result of a growing achievement gap between children with highly and less-educated parents. Indeed, the relationship between parental education and children’s achievement has remained relatively stable during the last fifty years, whereas the relationship between income and achievement has grown sharply. Family income is now nearly as strong as parental education in predicting children’s achievement.
Long story short, we're underinvesting as a society in early childhool education and other things that could help boost the fortunes of lower-income kids. Meanwhile, our K-12 system fails to narrow pre-existing gaps at all, which is a huge second missed opportunity given that we are dedicating a lot of financial resources to K-12 schooling.