An expression for the mutual diffusion coefficient of electrolyte solutions on the Smoluchowski level is proposed and is used in Brownian dynamics simulations. In particular, the divergence of this diffusion coefficient due to the long range of hydrodynamic interactions is solved by introducing explicitly the solvent-fixed frame of reference. The result is used to compute the mutual diffusion coefficient of KCl and LiCl aqueous solutions in the framework of the primitive model at room temperature. Simulations with and without hydrodynamic interactions were performed. Results are in excellent agreement with those obtained by an analytical transport theory which was recently proposed [J.-F. Dufreche , J. Chem. Phys. 116, 2085 (2001)] and which yields reliable estimates for transport coefficients. Moreover, it is shown that electrostatic relaxation effects which cancel exactly in the analytical theory have indeed little influence on the mutual diffusion.