This prestigious award was given for representing the Unseen University at a high-level sporting event. Both Mustrum Ridcully and the previous Dean were awarded Browns for Rowing in their younger days. As the river Ankh is no limpid sweet-flowing Isis, the award of a Brown to members of the university's boating team is entirely appropriate.
Actually, the Boat Race itself is a running race across the crust of the Ankh carrying the boat...
The Ankh being what it is, the boots generally corrode rather badly, another reason for their brown-ness... Ridcully still displays a pair of mouldering brown boots on the walls of his study in recognition of his meritorious holding of a Brown.
It is possible that the qualifier of "representing the Unseen University at a high-level sporting event" would also apply to sports such as Foot-The-Ball. In which case all members of the university's team in Unseen Academicals received an official Brown, including Ponder Stibbons, the Librarian, and Trevor Likely.
A parody of the "Blues" awarded at British and Australian Universities on Roundworld. A Blue is an award earned by sportsmen and women at a university for competition at the highest level. The awarding of Blues began at Oxford and Cambridge Universities in the late 1830s.