The erosion of the safe basins of a softening Duffing oscillator under harmonic or bounded random noise is studied in detail using the Monte-Carlo method. As an alternative, stochastic bifurcation may be defined as a sudden change in the character of the stochastic safe basin when the bifurcation parameter of the system passes through a critical value. This definition applies equally well either to randomly perturbed motion, or to purely deterministic motion. It is found that random noise may destroy the integrity of safe basins and bring about stochastic bifurcation, hence make the system more unsafe.