I admit to uncertainty about the proper resolution of this issue. The case against Harvard's decision:
We should make it as easy as possible for people to repudiate past bigotry and seek integration into polite society, assuming their contrition is sincere (as seems to be the case here). This is less for their benefit, than for that of the larger objective of reducing racism, and giving people incentives to reject it.
Kashuv apologised (apparently before this issue endangered his Harvard admissions chances).
As many have pointed out, he said this when he was 16, and many people say awful things at that age, which should not tar them for life.
There may well be an ideological double standard here. Had Kashuv made comparably awful left-wing statements and they were publicised, I am far from convinced Harvard would have rescinded the admissions offer. Consider, for example, if he had praised Mao's Cultural Revolution and urged other nations to imitate it.
But the opposite case is at least equally good:
Pretty much all selective college admissions policies involve judging people based on youthful mistakes they might not repeat after they mature into adults. Get bad grades for a semester or two because you slacked off or became depressed (this actually happened in my case)? Admissions offices will probably penalise you for it. Have a disciplinary offence as a sophomore that might not reflect your behaviour when you are 18? Ditto.
Statements made at the age of 16 are much more probative for assessing the character of an 18 year old college applicant than, say, a 38 year old, or a 58 year old. In the latter cases, there is track record of many more recent years to judge you by. In the former, your high school record is literally all we have (setting aside pre-HS behaviour, which is really that of a child).
There is a big difference between concluding that Kashuv should be permanently ostracised from polite society for these racist statements, and merely concluding they should disqualify him from going to Harvard - one of the world's most selective universities, which likely has many rejected applicants with academic records comparable to his, and zero evidence of moral taint. In all likelihood, Kashuv will still get to go to a highly selective college, just one somewhat less prestigious than Harvard.
What should Harvard have done? I am not sure.
Reinstate Kashuv, but only on condition that he first, say, spend a year in a job that involves combating racial or ethnic discrimination.Such a job need not involve working for a left-wing group. There are a number of conservative, libertarian, and non-ideological organisations involved in such work, too.
Stick to its decision on Kashuv, but make a credible commitment to clearly explaining its principles in this regard, and outlining them in a way that minimises ideological bias. If racist remarks are enough to get you rejected, the same should go for promoting comparably despicable left-wing ideas. This would require what may be a painful discussion of what kinds of ideas qualify as beyond the pale for an institution otherwise committed to free inquiry.
Conclude that Harvard lacks the institutional competence to fairly conduct the sort of evaluation required by option 2, and announce that, from now on, applicants' views on political and social issues will not be considered in admissions decisions. The price of this would be letting in a few people with excellent academic and extracurricular records, but also awful views. But it would eliminate ideological bias in admissions, which is otherwise hard to police.