A straightforward question. The answer is a bit more complex. Sorry if this is too long or gets in the weeds.
I look at our constitution as a document that covers how our government works, the powers it has, the relationship between the states and fed, and the powers it doesn't have...including, but not limited to the BoR,...in other words the fed's relationship with us.
In that context, what the fed is allowed to do is outlined as duties, mandates, requirements, etc. The function, powers, and relationship with states are not outlined as prohibitions. They are affirmed or stated. The powers it doesn't have, or said another way, the power (rights) that we have are outlined not as mandates or affirmed. They are written as prohibitions on the fed.
The amendments are a little bit of both approaches. Sometimes with good results. Sometimes not.
Lastly, I believe what isn't expressly outlined as a responsibility of the federal defaults to the states.
With that general overview, my answer is: there is no prohibition on the federal funding higher education because the document isn't structured that way. The structure would (or should) require that duty to be affirmed. And since higher education is under the control of states, the federal should not have any duty. It isn't affirmed and it defaults to states in application as well as controlled by the states in responsibility.