We study the dynamics of charged particles in aqueous solutions in the framework of the continuous solvent model by smart Brownian dynamics simulations. Transport coefficients of simple electrolyte solutions (KCl between 0.1 and 2 mol/L) are computed from the simulations and compared to experimental data. The simulation method is then applied to more complex systems : aqueous cryptate solutions. In the latter case, we compare the experimental values for the conductivity to the simulation results. Both direct and indirect hydrodynamic interactions are taken into account in the simulations : the first ones are modelled by a solvent-averaged pairwise interaction potential and the second ones are evaluated by using the Rotne-Prager approximation. The smart Brownian dynamics method allows to use long time steps of about 0.1 ps and to generate trajectories of several ns in total. It is shown that hydrodynamic interactions are crucial to compute transport coefficients in agreement with experimental data. The self-diffusion coefficients are slightly enhanced and the conductivity is lowered when hydrodynamic interactions are taken into account.