This paper studies constrained linear-quadratic regulator (LQR) problems in distributed boundary control systems governed by the Stokes equation with point velocity observations. Although the objective function is not well defined, we are able to use hydrostatic potential theory and a variational inequality in a Banach space setting to derive a first-order optimality condition and then a characterization formula of the optimal control. Since matrix-valued singularities appear in the optimal control, a singularity decomposition formula is also established, with which the nature of the singularities is clearly exhibited. It is found that in general, the optimal control is not defined at observation points. A necessary and sufficient condition that the optimal control is defined at observation points is then proved.