Within the Monte Carlo formalism supplemented by the modified Newton-Raphson optimization technique, we investigated structural and dynamical properties of two-dimensional binary clusters confined in an external hard-wall potential. Two species of differently charged classical particles, interacting through the repulsive Coulomb force are confined in the cluster. Subtle changes in the energy landscape and the stable cluster configurations are investigated as a function of the total number of particles and the relative number of each of the two particle species. The excitation spectrum and the normal modes corresponding to the ground-state configuration of the system are discussed, and the lowest nonzero eigenfrequency as a measure of the stability of the cluster is analyzed. The influence of the particle mass on the eigenfrequencies and eigenmodes are studied, i.e., we study a binary system of particles with different charge and different mass. Several unique features distinct from a monodisperse system are obtained.