The dissipation of the structural vibration energy at viscoelastic supports is an efficient method of reducing modal resonances and consequent noise and fatigue related problems. The support stiffness has significant impact on the modal characteristics. The dissipation capabilities of the viscoelastic support depend on its stiffness. Methods to optimally tune this support stiffness are proposed in this study. The characteristic mechanical impedance for structural vibration is obtained from wave propagation analysis and non-reflecting boundary conditions. The wave propagation is analyzed near the supports installed at edges, middle of a structure, and for the tuned vibration absorber. The dependence of the optimal stiffness on the location and mass of the supports is identified. A simple analytical solution for optimal support stiffness for maximum dissipation of propagating vibration energy at supports is presented.