The erosion of the safe basins and related chaotic motions of a softening Duffing oscillator under harmonic and bounded random noise are studied. By the Melnikov method, the system's Melnikov integral is computed and the parametric threshold for the onset of chaos is obtained. Using the Monte-Carlo and Runge-Kutta method, the erosion of safe basins is also discussed. As an alternative definition, stochastic bifurcation may be defined as a sudden change in the character of stochastic safe basins when the bifurcation parameter of the system passes through a critical value. This definition applies equally well to either randomly perturbed motions or purely deterministic motions. It is found that random noise may destroy the integrity of the safe basins, bringing forward the stochastic bifurcation and making the threshold for onset of chaos vary to a large extent, which makes the system less safe and chaotic motion easier to occur.