Domain optimization problems for the two-dimensional stationary Stokes equations are studied. Frechet differentiability of a class of cost functionals with respect to the variation of the shape of the computational domain is established. An embedding domain technique provides an equivalent formulation of the problem on a fixed domain and, moreover, a simply computable formula for the derivative of the cost functional with respect to the domain. Existence of a solution to the class of domain optimization problems is proved. Numerical examples show the reliability of the derivative formula.