An asymmetric rotor in aerospace vehicle is usually subjected to random axial loads by reason of the aerodynamic forces, hydrodynamic forces, preloads and so on. The modeling of an asymmetric rotor subjected to stochastical axial loads is proposed as stochastically excited and dissipated Hamiltonian systems. Two cases, (1) the asymmetric rotor with the external damping and (2) the asymmetric rotor with the internal and external damping, are studied with the aid of the stochastic averaging method for quasi- integrable-Hamiltonian systems. The necessary and sufficient conditions for the almost sure asymptotic stability of the asymmetric rotor are presented approximately and the expression for the largest Lyapunov exponent of the averaged equations is formulated for investigating the stochastic stability of the asymmetric rotor under random axial loads. It is found that the ratio of the external damping coefficient to the rotor mass plays a key stabilizing role in the almost sure stability when the external damping of the asymmetric rotor is taken into account. Considering both the external and internal damping, the effects of stochastical axial loads on the almost sure instability are significant. The influence of the spectral density on the almost sure stability of the asymmetric rotor with both the external and internal damping is reduced, which is different from that with only the external damping. The effects of the flexural rigidity variation on the almost sure instability of the asymmetric rotor with the external damping are more considerable than that with both the external and internal damping.