In this series of two papers we investigate phase instabilities in charged hard-sphere mixtures. Here, we derive general expressions for the thermodynamic properties of a common anion mixture and apply these to study phase separation in mixtures of salt and hard spheres. Excess thermodynamic properties due to Coulombic interactions are obtained using the analytical solutions for the mean spherical approximation closure. A detailed description of the dependence of the resulting phase diagrams on charge asymmetry of the ions, size of the neutral species, and the osmotic pressure of the mixture is presented. Binary mixtures of salt and hard spheres exhibit type III phase behavior. An increase in charge asymmetry results in an increase in the critical temperature of the mixture because enthalpic forces (ion-pairing) dominate. An increase in the size of the neutral species also results in an increase in the critical temperature of the mixture because of packing effects, which encourage phase separation. Potential applications of the model to experimental systems are discussed.