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A new type of Brownian motion, stochastic mass, has received widespread attention and has had a significant impact on the dynamics of the systems with light mass and small volume. This paper investigates the asymptotic stability with probability one of a nonlinear energy harvester with mass disturbance undergoing Markovian jump. To simplify the mass disturbance, which is modeled as Gaussian white noise, an approximate non-dimensional system is established. Using the stochastic averaging method, the averaged stochastic differential equation of the amplitude in the approximate system is derived. With the linearization,we deduced the largest Lyapunov exponent of the linearized equation. Furthermore, the necessary and sufficient condition for asymptotic stability of the nonlinear energy harvester is proposed approximately by managing the largest Lyapunov exponent be negative. The effects of mass disturbance and Markovian jump on the asymptotic stability with probability one of the nonlinear energy harvester are analyzed. To verify the accuracy of the results, we calculate the displacements of the original system directly numerically. In addition, the stability regions of the nonlinear energy harvester with different transition matrixes and stochastic mass noise intensities are thoroughly studied.