This paper deals with the choice of stabilization parameter for the grad-div stabilization applied to the generalized Oseen equations. In particular, inf-sup stable conforming pairs of finite element are used to derive the stabilization parameter on the basis of minimizing the $$H^1({\varOmega })$$ H 1 ( Ω ) error of the velocity. For the proposed choice of the parameter, the $$H^1({\varOmega })$$ H 1 ( Ω ) error of the velocity is derived that shows a direct dependence on the viscosity coefficient. Differences and common features with the Stokes equations are discussed. Numerical studies are presented which confirm the theoretical results. Moreover, for the Navier-Stokes equations, numerical simulations are performed on a two-dimensional flow past a circular cylinder. It turns out that, for the MINI element, the best results are achieved without grad-div stabilization.