When social mobility is near non-existent, then class has effectively become it's very own immutable characteristic. People can be born into wealth and poverty just as much as they can be born with colored skin.
The "American dream" is the anti-thesis to that, stipulating that everybody can elevate themselves out of their lower-class if they just pull those bootstraps hard enough.
But how realistic is that actually nowadays, particularly to elevate larger parts of the population?
the current prominence of immigration in political discussions, for example, makes it easy to divide on racial lines rather than on fairness or economics, as republicans have effectively exploited. it's led to an economically irrational willingness to sacrifice growth (and its resultant opportunities) in favor of limiting non-white immigration (we cheer 2-3% growth as amazing, while other countries have double-digit growth--like china, whose economy is similarly sized).
so support immigration on economic grounds rather than racial ones, because it can grow the pie more fairly than plutocracy can. most white folks can get behind more growth, but not all will get behind direct compassion and fairness for non-white folks (as bad as that sounds).
...which is frequently deployed against people with a different skin color. Class-based analysis is great, but a) it is not an exact science, contrary to the claims of many of its proponents, and b) avoiding the empirical fact of racial discrimination does not make it go away, any more than ignoring the empirical facts of misogyny or any other flavor of discrimination has caused them to go away.
You can argue that these different forms of discrimination are just disguises for class-based discrimination, but it's equally arguable that discrimination against people on the basis of immutable characteristics is a highly effective class ordering mechanism.