New explicit stability conditions are derived for stochastic integro-differential equations with operator coefficients. The equations under consideration arise in the analysis of viscoelastic structural members driven by random excitations. Unlike previous studies, the coefficients are not assumed to be self-adjoint and commuting operators. Our stability conditions are formulated in terms of norms of the operator coefficients and some auxiliary operators, as well as four specific characteristics of kernels of the integral operators. The conditions developed are applied to determine the critical intensity of random load applied to a system of two viscoelastic bars linked by a glue layer. The effect of material and geometrical parameters of the system on the critical load is analyzed numerically