The first passage problem for the response of a linear oscillator excited by a random excitation is considered. An approximate analytical technique is presented for calculation of the distribution of the time to first excursion across a symmetric double barrier. The approach may be applied to the case of nonstationary response to modulated Gaussian noise with nonwhite spectral density. Results for the limiting decay rate parameter are presented and are compared with those of other analytical methods and simulation results. The first passage probability is calculated for a system subjected to a suddenly applied white noise and the results also compared with those of other methods and computer simulations. The results of the proposed method show generally good agreement with simulation results.