This paper studies the 2nd-moment exponential stability of a class of infinite-dimensional linear stochastic switched systems comprising two unstable subsystems. We first construct an algebraic sufficient condition on the existence of multiple Lyapunov functions. Then, two switching strategies are designed to stabilize infinite-dimensional linear stochastic switched systems in terms of the multiple Lyapunov function method. Moreover, the system possesses good robust stability of the switching time with our switching strategies.