In this note, a new type of stability theorem for stochastic systems is established firstly. Based on this stability theorem and its corollaries, stochastic stabilization and destabilization by noise are further investigated. In the note, the local Lipschitz condition is weakened to the generalized local Lipschitz condition. The commonly used linear growth condition or one-side linear growth condition is weakened to the generalized one-side linear growth condition, which is local, variable and nonlinear, admits nearly arbitrary variability in the time and real nonlinearity in the state. As an application, a simple and direct design method is proposed for finding a noise strength (g) over cap (t, x) so that the added noise (g) over cap (t, x) d (B) over cap (t) stabilizes an unstable stochastic system or destabilizes a stable one. A numerical example is presented at the end of the note to illustrate the usage and efficiency of the proposed design method of the note.