The Boat Race is a running race across the crust of the Ankh carrying the boat. The Compleat Ankh-Morpork goes into more detail and relates that "The Unseen University Boat Race is a good spectator sport, consisting as it does of two teams of students wearing very wide-soled shoes running across the river from the University boatyard to the Brass Bridge whilst carrying a racing skiff".
The account further notes that " It is possible to do this since the river Ankh is so turgid that it can barely be considered mobile. In the earlier days the rowing boats had a man at the helm to dig a way through the water. This is no longer required." (The Compleat Ankh-Morpork, p 93). This may add a little clarification to the individual award of "Head of the River" which is given in passing in the canonical text.
Browns are the prestigious award 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. 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.