Stochastic viral dynamics modeled by stochastic differential equations with Beddington-DeAngelis functional response and rates driven by white noise will be presented. The stochastic positive invariance and the existence of stationary distribution are proved. Through estimation of the pathwise and asymptotic moment upper bounds, the moment Lyapunov exponent is shown to be nonpositive when the noise intensities are relatively small, and the persistence and extinction will be discussed.