Masonry infills are generally assumed as non-structural elements in structural calculations and are not modeled. However, observations after past earthquakes have shown that masonry infills have significant effects on the seismic performance of structures and their seismic behavior should not be neglected. Additionally, the absence of masonry infills in the first story, which is common in structures for commercial and architectural reasons, has led to the occurrence of the soft story phenomenon in past earthquakes. The maximum interstory drift ratio (MD) is the most important criterion for assessing seismic damage and the occurrence of collapse in structures. In this study, the seismic performance of 3- and 9-story steel moment resisting frames (MRFs) with masonry infills was evaluated using a probabilistic framework considering the record-to-record variability. Two configurations were considered for the masonry infills including fully infilled and open ground story configurations. The seismic performance of the MRFs with these two configurations was compared to that of bare MRFs. The OpenSees software was employed for nonlinear modeling of the structures and masonry infills were modeled using single compression-only struts. The fundamental periods of structures with masonry infills significantly increase after the failure of the masonry infills. To evaluate these effects, Rayleigh damping was modeled using the conventional method and a modified method, which considers the severe elongation of fundamental period due the failure of infills, and the responses obtained from the two methods were compared. By performing incremental dynamic analyses using 78 far-field ground motion records, drift margin ratios (DMRs), drift fragility curves and mean annual frequencies of exceeding four MD levels of 0.7%, 2.5%, 5%, and 15% (λ_MD) were obtained for the structures. The MD levels of 0.7%, 2.5%, and 5% correspond to the performance levels of immediate occupancy, life safety, and collapse prevention, respectively. The MD level of 15% corresponds to the seismic collapse of the structures. The results indicate that the presence of masonry infills improves the drift performance of the MRFs with the fully infilled configuration. However, since the masonry infills experience failure at higher drift levels, their effectiveness decreases at these drift levels. Furthermore, the absence of masonry infills in the first story leads to the soft story phenomenon at lower drift levels, and therefore, the performance of the structures with the open ground story configuration is worse than that with the fully infilled configuration. It should be mentioned that at higher drift levels, due to the failure of masonry infills, the structures with the two configurations for infills have almost the same performance and close to the performance of the bare MRFs. The amplification of the soft story phenomenon in the 9-story structure causes the performance of the structure with the open ground story configuration given some drift levels to be even worse than that of the bare structure. For example, by using the modified damping method for the 3-story structure with the fully infilled configuration, the masonry infills reduce the λ_MD value given MD = 0.7% by 43%, but the reduction in the λ_MD given MD = 15% is 19%. Based on the results obtained, the conventional Rayleigh damping method in the technical literature underestimates the responses.