The use of homing-based gene drive systems to modify or suppress wild populations of a given species has been proposed as a solution to a number of significant ecological and public health-related problems, including the control of mosquito-borne diseases. The recent development of a CRISPR-Cas9-based homing system for the suppression of Anopheles gambiae, the main African malaria vector, is encouraging for this approach; however, with current designs, the slow emergence of homing-resistant alleles is expected to result in suppressed populations rapidly rebounding, as homing-resistant alleles have a significant fitness advantage over functional, population-suppressing homing alleles. To explore this concern, we develop a mathematical model to estimate tolerable rates of homing-resistant allele generation to suppress a wild population of a given size. Our results suggest that, to achieve meaningful population suppression, tolerable rates of resistance allele generation are orders of magnitude smaller than those observed for current designs for CRISPR-Cas9-based homing systems. To remedy this, we propose a homing system architecture in which guide RNAs (gRNAs) are multiplexed, increasing the effective homing rate and decreasing the effective resistant allele generation rate. Modeling results suggest that the size of the population that can be suppressed increases exponentially with the number of multiplexed gRNAs and that, with six multiplexed gRNAs, a mosquito species could potentially be suppressed on a continental scale. We also demonstrate successful multiplexing in vivo in Drosophila melanogaster using a ribozyme-gRNA-ribozyme (RGR) approach - a strategy that could readily be adapted to engineer stable, homing-based suppression drives in relevant organisms.