I knew some college students in the U.S. who were <18 years old.
In which countries must college professors be cleared (background-checked, certified, fingerprinted, etc., like U.S. K-12 teachers) to work with minors?
Any special requirements or background checks regarding working with minors are a matter of employer policy or local law, and there are no standard rules across different jurisdictions. In particular, there's no way to give a definitive answer without knowing the exact circumstances.
As a general rule, though, these sorts of background checks are rarely applied to college faculty. Even in locations with unusually strict laws, there are often exceptions for college professors, despite the fact that some students may be under 18. For example, the University of Sydney policy explicitly says "Where University staff or affiliates have direct contact with University students under the age of 18, this is not regarded as child-related work under the Act."
I'm not aware of any countries where background checks are required specifically to work with minors. There are places where background checks occur as a general condition of employment, but it's just because everybody gets a background check, and not just because they could work with minors.
As AM points out, in general, minors enrolled in universities are treated like everybody else, so there's generally no reason why there would be special background checks.
A quick google gave me several US university that have policies on the issue. For example Georgetown's policy says staff involved in programs involving minors should do some training and that supervisors of those programs and those that regularly spend time alone with minors have to have a criminal records check.
I think the situation in the UK is similar with staff in one to one contact or on programs aimed at minors having to do a DBS/CRB check. See Imperial's policy.
I'm not sure but I expect the situation in at least western Europe is similar.