AIDS is an extremely harmful infectious disease since HIV virus destroy the immune system of people. Homosexuality is one of the important ways to spread HIV. Considering the transmission characteristics of men who have sex with men (MSM) and the existence of random factors in the environment, a stochastic HIV/AIDS epidemic dynamic model based on MSM is established. Using the basic theory of stochastic differential equation, the dynamic behaviors of the system are analyzed. First, for any given positive initial value, it is proved that there is a unique and global positive solution. From a biological point of view, this conclusion must hold, which ensure it is value of studying such a system. When a virus dynamical model is investigated, the extinction and persistence of virus require the immediate attention. Therefore, by using It$\hat{\rm o}$'s formula and constructing some special functions, the sufficient conditions of disease extinction are given. Moreover, the existence of a unique positive ergodic stationary distribution of the system is discussed, which implies the prevalence of disease. Finally, the obtained theoretical results are verified by numerical simulation, and the influences of random disturbance on the dynamical behavior of the system are analyzed.