Probability theory offers an appropriate framework for quantifying the level of safety of a structural system. Thus, it is possible to evaluate the chances that a structure fulfills prescribed performance criteria during its lifetime due to uncertainties in loadings, deterioration processes, etc. The aforementioned chances are usually termed as the reliability associated with a structure. Although reliability constitutes a fundamental metric of safety, knowledge on the sensitivity of the reliability with respect to variables such as material properties, cross section of structural members, etc. is also of paramount importance. The information on reliability sensitivity is most valuable as it allows, for example, taking decisions on the final design of a structure such that its level of safety is relatively invariant in case unexpected changes occur. Within the aforementioned framework, this contribution presents a novel approach for estimating the reliability sensitivity of a particular type of structures, namely linear structural systems subject to dynamic loading modeled as a Gaussian stochastic process. The sensitivity is computed with respect to deterministic variables that affect the structural performance (such as stiffness, cross section of structural members, etc.). The proposed approach is based on an efficient simulation technique and local approximations of the functions used to model the structural performance. The reliability measure is expressed in terms of the first excursion probability, which is a criterion widely used in stochastic structural dynamics. In addition, the error associated with the estimator of the reliability sensitivity is computed by means of the Bootstrap method.