One thing I don't want to get into is whether or not the university did enough, or didn't do enough or whatever in preventing of not preventing the violence. That would be a set of facts for a jury to decide, and a separate question. Let's just assume the intent of the university was to ban Milo Yiannopoulos all along. (I don't think that's the case).
There are a few ways then blocking funds potentially could be done. The first is the Solomon Amendment, but note some conditions may need to exist. However, Berkeley has both a ROTC program and military recruitment at its law school, which is where the tensions usually bubble up. Generally speaking there has been much reluctance to enforce the Solomon Amendment by GOP presidents and very little if any by Democratic ones.
The 1996 Solomon Amendment is the popular name of 10 U.S.C. § 983, a United States federal law that allows the Secretary of Defense to deny federal grants (including research grants) to institutions of higher education if they prohibit or prevent ROTC or military recruitment on campus.
But that doesn't necessarily apply here. However, the Office of Civil Rights(OCR) at the Department of Education certainly does. There are a number of fairly easy arguments that a Trump Administration could make that Berkeley violated civil rights:
The Office for Civil Rights enforces several Federal civil rights laws that prohibit discrimination in programs or activities that receive federal financial assistance from the Department of Education. Discrimination on the basis of race, color, and national origin is prohibited by Title VI of the Civil Rights Act of 1964; sex discrimination is prohibited by Title IX of the Education Amendments of 1972; discrimination on the basis of disability is prohibited by Section 504 of the Rehabilitation Act of 1973; and age discrimination is prohibited by the Age Discrimination Act of 1975. These civil rights laws enforced by OCR extend to all state education agencies, elementary and secondary school systems, colleges and universities, vocational schools, proprietary schools, state vocational rehabilitation agencies, libraries, and museums that receive U.S. Department of Education funds. Areas covered may include, but are not limited to: admissions, recruitment, financial aid, academic programs, student treatment and services, counseling and guidance, discipline, classroom assignment, grading, vocational education, recreation, physical education, athletics, housing, and employment. OCR also has responsibilities under Title II of the Americans with Disabilities Act of 1990 (prohibiting disability discrimination by public entities, whether or not they receive federal financial assistance). In addition, as of January 8, 2002, OCR enforces the Boy Scouts of America Equal Access Act (Section 9525 of the Elementary and Secondary Education Act of 1965, as amended by the No Child Left Behind Act of 2001). Under the Boy Scouts of America Equal Access Act, no public elementary school or State or local education agency that provides an opportunity for one or more outside youth or community groups to meet on school premises or in school facilities before or after school hours shall deny equal access or a fair opportunity to meet to, or discriminate against, any group officially affiliated with the Boy Scouts of America, or any other youth group listed in Title 36 of the United States Code as a patriotic society.
At first reading, you may say Milo Yiannopoulos is not a protected class. Ideology is not a protected class. However, it's pretty easy to extend Yiannopoulos's views to be pro-male. And Title IX is what you have to worry about, because there is a whole set of precedents and apparatus in place to extend regulations through the use of Dear Colleague letters.
Civil rights regulation contains few clear-cut, publicly proclaimed rules. It rests instead on layer after layer of administrative guidelines, interpretive memos, suggestions included in enforcement handbooks, judicial interpretations of statutes and of agency rules, and even more esoteric judicial doctrines on burden of proof. The sources of these multiple rules and standards are often obscure. Agencies claim to rely on the authority of courts; courts claim to rely on the expertise of administrators; and both judges and administrators claim to follow the commands of Congress.
This is what the folks at Powerlineblog had to say about the ease of abusing Dear Colleague letters:
The thing to note here is that the Clinton Administration’s “guidance” on Title IX was what is known in the regulatory trade today as an “informal” rule-making: the regulatory regime does not go through the normal Federal Register notice, public comment period, and judicial review that is intended by the Administrative Procedure Act. This is not an accident: activist liberals know that many of the regulations they want to foist in the world wouldn’t survive this formal process. So they rely on “Dear Colleague” letters that say, “If you don’t do what we say, we’ll investigate you for civil rights violations, and possibly cut off your federal funds.” The Obama Administration ran the same drill with a “Dear Colleague” letter demanding that universities set up investigations of harassment and intimidation claims with a lower evidentiary standard—i.e., without proper due process rights.
Finally, the President could negotiate through the budget process restrictions for Berkeley directly, direct other agencies (Dept of Energy and Defense come readily to mind) contracts and grants to other schools, and perhaps others.