We develop the BRST approach for all massless integer and half-integer higher spins in 4D Minkowski space, using the two component spinor notation and develop the Lagrangian formulation for supersymmetric higher spin models. It is shown that the problem of second class constraints disappears and the BRST procedure becomes much more simple than in tensorial notation. Furthermore, we demonstrate that the BRST procedure automatically provides extra auxiliary components that belong in the set of supersymmetry auxiliary components. Finally, we demonstrate how supersymmetry transformations transformations are realized in such an approach. As a result, we conclude that the BRST approach to higher spin supersymmetric theories allows to derive both the Lagrangian and the supersymmetry transformations. Although most part of the work is devoted to massless component supersymmetric models, we also discuss generalization for massive component supersymmetric models and for superfield models.