Understanding the pleiotropic consequences of gene drive systems on host fitness is essential to predict their spread through a host population. Here we study Sex-ratio (SR) X-chromosome drive in the fly Drosophila recens, where SR causes the death of Y-bearing sperm in male carriers. SR males only sire daughters, which all carry SR, thus giving the chromosome a transmission advantage. The prevalence of the SR chromosome appears stable, suggesting pleiotropic costs. It was previously shown that females homozygous for SR are sterile, and here we test for additional fitness costs of SR. We find that females heterozygous for SR have reduced fecundity and that male SR carriers have reduced fertility in conditions of sperm competition. We then use our fitness estimates to parameterize theoretical models of SR drive and show that the decrease in fecundity and sperm competition performance can account for the observed prevalence of SR in natural populations. In addition, we find that the expected equilibrium frequency of the SR chromosome is particularly sensitive to the degree of multiple mating and performance in sperm competition. Together our data suggest that the mating system of the organism should be carefully considered during the development of gene drive systems.