Numerical stability of both explicit and implicit Runge-Kutta methods for solving ordinary differential equations with an additive noise term is studied. The concept of numerical stability of deterministic schemes is extended to the stochastic case, and a stochastic analogue of Dahlquist's A-stability is proposed. It is shown that the discretization of the drift term alone controls the A-stability of the whole scheme. The quantitative effect of implicitness upon A-stability is also investigated, and stability regions are given for a family of implicit Runge-Kutta methods with optimal order of convergence.