Discontinuous Galerkin (DG) methods are a class of efficient tools for solving fluid flow problems. There are in the literature many greatly successful DG methods. In this paper, a new staggered DG method for the Stokes system is developed and analyzed. The key feature of our method is that the discrete system preserves the structures of the continuous problem, which results from the use of our new staggered DG spaces. This also provides local and global conservation properties, which are desirable for fluid flow applications. The method is based on the first order mixed formulation involving pressure, velocity, and velocity gradient. The velocity and velocity gradient are approximated by polynomials of the same degree while the choice of polynomial degree for pressure is flexible, namely, the approximation degree for pressure can be chosen as either that of velocity or one degree lower than that of velocity. In any