This is referring to the U.S.
I know that schools like Yale and Princeton used to dominate college football and basketball, among other sports, in the 19th and early 20th centuries. It seems like a shift occurred somewhere in the mid-late 20th century to big-time public university programs, at least in all the "major" sports. The only schools I can think of that are private, academically top-tier, and competitive at major sports are Stanford, USC, and Notre Dame. Today, the major sports are all pretty much dominated by The University of X State.
When exactly did this shift occur, and why? Is it simply a function of Ivy Leagues not giving sports scholarships? If so, why didn't other private schools take up the mantle, or why did big, state schools take it up instead?
This might not be a great, source-able topic for AskHistorians.
That said, if you want to look at the shift here, there are probably three simple reasons, and I don't think they have a direct correlation with the divide between public and private.
Offering sports scholarships will increase the attractiveness of sports programs to elite athletes. Those schools that don't include the Ivy League, which has a lot of the formerly dominant programs (Princeton for Football, for example). But not all private schools have restrictions on sports scholarships like those in the Ivy League, so this is not a strictly Public/Private issue.
High academic expectations/entry requirements will narrow the field of eligible elite athletes, and the Ivy League schools all have high barriers to entry. Again, this is not strictly a public/private issue though. There are some public schools with very high entry/grade requirements, and some of them still have very successful athletic programs (though making exceptions for student athletes is a common theme at big sports schools).
Budgets for athletics. A lot of the biggest schools in the major sports (Basketball and Football) have enormous budgets for their programs. Better facilities and better coaching allow teams to attract more talent and train that talent more effectively. Some private schools have chosen not to follow the same pattern of sports investment.
It may be important to note that, if you look at athletics outside of the major sports, Ivy League teams can often be powerhouses. Rowing/Crew, Lacrosse, Squash, Field Hockey, etc. teams at the Ivy schools are often excellent, but they also benefit from relatively high levels of funding at those schools.