Based mostly on what is reported on http://www.collegedata.com , white undergraduate percentage of 45.0% or lower among prominent selective schools frequently discussed on these forums:
(private schools)
45.0% Wellesley
44.6% Rice
44.6% NYU
43.4% USC
41.0% MIT
40.4% Cooper Union
40.3% Stanford
38.4% Columbia
33.4% Caltech
(public schools)
42.0% Stony Brook
32.4% UCLA
24.9% Berkeley
However, it is the case that many of the most selective colleges, including most of the Ivy League, are right at around 50% white undergraduate enrollment. If, as you say, that many white students hesitate to attend a school where they would be members of an ethnic minority group, then perhaps it would not be surprising if many of these colleges (who get so many well qualified applicants that they can build their classes however they like) prefer to have white enrollment at that level in order to avoid a marketing problem to white students while having a high enough enrollment of each non-white group to improve marketing to non-white students who, as you say, may not want to feel isolated.