The stability of ionic binary mixtures is investigated by an integral equation method. In presenting the theory a distinction is made between primary and secondary stability criteria, and this distinction is used to clarify some misconceptions in the literature. The derived stability criteria are then applied to electrolyte solutions as well as to a simple binary mixture. In a simple mixture of hard spheres in wafer, both mechanical and material instabilities are found near the spinodal line along with evidence of long-range hydrophobic forces. Results for the electrolyte solutions indicate that salts with only large ions, such as CsI, and those with a smaller ion, such as Na+ or K+, behave differently near the spinodal line. CsI acts hydrophobicly, and appears to undergo demixing from the solvent, whereas NaCl and KCl, which bind the solvent more tightly, do not show clear signs of any such demixing, but do appear to become mechanically unstable. Finally, some recent results of Chen and Forstmann [J. Chem; Phys. 97, 3696 (1992)] are discussed and applied to the present systems.