Many private universities in the US, including Harvard, have a reputation that is built not only (or even primarily) on the quality of their undergraduate education but on the quality of their research output. The funding for this research, which includes compensation for professors, grad students, and operational overhead, forms a significant component of the revenue of the institution overall. It frequently dwarfs undergraduate tuition revenue and often constitutes the largest single component of institutional revenue.

A substantial portion of this funding is federal. It's in the public interest for the positive effects of being educated at a well-funded research institution to be allocated "fairly" and since that funding comes from public sources it is well within the purview of the government to regulate that allocation.