1. As I understand it, the traditional, essentialist notion of 'race' is outmoded: there are no classes of beings that have 'blackness,' 'whiteness,' or what have you as an essence that defines them. There is often more genetic variation between members of the same race than there is between members of different races. (Another way of putting the same point: there isn't any property that all and only white people have, and so goes it for all commonsense racial groupings.)
2. If we want to construct a more subjective notion, we can conjoin the various light-skinned ethnicities and call this grouping the 'white race,' as long as we recognize that the grouping is mostly arbitrary, rather fluid, subject to change over generations, and necessarily a victim of vagueness.
3. Once we've done this, we see that the fate of the white race is determined in whole by the fate of the sub-groups we've gerrymandered together, plus the particular criteria of success we decide upon:
Criterion 1: ESF (Eat, sleep, fuck). Whites will continue to eat, sleep, and fuck at sufficient rates to survive in the barest sense of the word. The desperately poor 'Appachalian white' sub-group will take up this task and attempt to be (at the very least) competitive with the brownies.
Criterion 2: Political influence. It's true that demographically speaking, the country is slated to become a sea of brown. The brownies will presumably overcome voter suppression meant to keep them out of the political process and thus, pending an ideological shift, will ordain a new era of Democratic Party dominance. That isn't necessarily bad for Jews or WASPs, most of whom are (or lean) Democrat anyway, but vis-a-vis brownies they will lose influence within the dominant party at the grassroots level. The desperately poor Appalachian whites, meanwhile, will have much less representation because the Republican Party will be a consistent loser.
Criterion 3: Wealth. Pending a miraculous shift in educational achievement, cultural practices, and, in particular, admissions criteria at the Ivy Leagues, WASPs and Jews will continue to utterly dominate in this category. This is because elite schools are funnels to top jobs in politics, business, finance, and much else besides, and Jews and WASPs are severely overrepresented at such schools. Cultural practices relating to handling money are heavily weighted in white people's favor, with only 2% of African Americans making investments early in their lives (the percent is very low for other minorities as well; the percent for whites is 20%). Also, since wealth has a profound impact in our political system (the affordances to Wall Street and NASCAR in the latest compromise over the deficit serve as a recent example), continued white dominance of this criterion bleeds over into criterion 2, somewhat counteracting the grassroots dominance that brownies will enjoy. Appalachian whites will stay as dumb and religious (almost coextensive concepts) as before.
So there you have it: save an unlikely confluence of factors (minority cultures suddenly inculcate success-inducing practices, elite schools change their admissions criteria, minorities suddenly start excelling at academics at an increased rate, wealth suddenly becomes irrelevant to influencing the political system), whites will not only survive, but will continue to dominate in virtue of two dominant sub-groups maintaining the reigns of power and a lesser, bumfuck sub-group bolstering white numbers by procreating overmuch.