We propose a novel passivity-based stabilization control framework for a certain class of nonholonomic mechanical systems. First, we derive the notion of passive configuration decomposition, which enables us to configuration-level decompose the system's Lagrange-D'Alembert dynamics into two systems, each evolving on their respective configuration spaces and also individually inheriting Lagrangian structure and passivity from the original dynamics. We then propose passivity-based time-varying and passivity-based switching control schemes that, by utilizing some control actions defined on each of the configuration spaces of the two systems and also their interplay with the nonholonomic constraint, can achieve stabilization while exploiting the nonlinear dynamics of the decomposed dynamics. We also provide conditions under which the passive configuration decomposition is possible, establish an equivalence between certain conditions for the proposed controls and kinematic controllability, and elucidate robustness property of the proposed controls based on passivity. Finally, examples are provided along with their simulation and experimental results to illustrate the theory.