We study the approximate controllability of a stationary Stokes system with linearized convection in a bounded domain of ℝN. The control acts on a part of the boundary and the velocity field is observed on an interior curve (N=2) or surface (N=3). We establish the L2-approximate controllability under certain compatibility conditions and suitable geometrical assumptions on the curve or surface. We build controls of minimal L2-norm by duality. To compute the control, we propose a numerical method, based on duality techniques, consisting in the minimization of a nonquadratic functional coupled to a Stokes system. It is tested in several situations leading to interesting numerical results.