This article studies the effect of a two-component plasma (hot and cold) on the shear driven Kevin--Helmholtz instability. The theory incorporates an ion distribution function with a shear flow parallel to the ambient magnetic field and a density gradient perpendicular to both. The electrostatic and electromagnetic modes of the instability are studied in the limit of hydromagnetic frequencies. The dispersion relation for the electrostatic case is obtained by solving Vlasov equation for the perturbed ion and electron densities that are then used in the quasi-neutrality condition. The electromagnetic dispersion relation is obtained by solving the coupled Vlasov and Maxwell equations. For low &bgr;( = 4&pgr;NkT/B2), the electromagnetic analysis yields two uncoupled modes. One is related to the Alfv¿n mode and the other to the ion acoustic mode. The Alfv¿n mode exists only in a finite &bgr; plasma, while in the limit &bgr;→0, the ion acoustic mode becomes purely electrostatic. The effect of cold plasma on the two modes is studied for a wide variety of parameters (velocity shear, density gradient, wave number, plasma &bgr;, and flow velocity). The results show that the ion acoustic mode is stabilized by the presence of very small amounts of cold plasma. For &bgr;≪1, the Alfv¿n mode is found to dominate over the ion acoustic mode when a sufficiently large background of cold plasma is present. The results applied to the magnetosphere show that the ionosphere plays an important role in determining the type of waves that are generated by a sheared plasma flow and indicate whether the flow boundary is unstable to the Kelvin--Helmholtz instability.