Vibration energy harvesters convert the energy of ambient, random vibration into electrical power often using piezoelectric transduction. The stochastic dynamics of a piezoelectric harvester with parameteric uncertainties is yet to be fully explored in the nonequilibrium regime. Motivated by mathematical results that establish the counterintuitive phenomenon of stabilization of response in certain nonlinear systems using noise, we investigate the stochastic stability of a generic harvester in the linear and the monostable nonlinear regimes excited by multiplicative noise characterized by both Brownian and the Lévy stable distributions. First, a lower bound on the magnitude of noise intensity that guarantees exponential stability almost surely, is obtained analytically as an inequality in terms of system parameters in the linear case. This result is validated numerically using the Euler-Maruyama scheme. Next, noise-induced stabilization in the harvester dynamics is demonstrated numerically for both the linear and nonlinear cases wherein Lévy noise was found to achieve stabilization at lower noise intensities than Brownian noise. Additionally, the inclusion of a nonlinear stiffness does not have an appreciable affect on the stabilization behavior. The results indicate that stabilization may be achieved using noise and are expected to be useful in harvester design.