We study by mean-field analysis and stochastic simulations chemical models for genetic toggle switches formed from pairs of genes that mutually repress each other. To determine the stability of the genetic switches, we make a connection with reactive flux theory and transition state theory. The switch stability is characterized by a well-defined lifetime t. We find that t grows exponentially with the mean number N of transcription factor molecules involved in the switching. In the regime accessible to direct numerical simulations, the growth law is well-characterized by t ~ Na exp(bN), where a and b are parameters. The switch stability is decreased by phenomena that increase the noise in gene expression, such as the production of multiple copies of a protein from a single mRNA transcript (shot noise) and fluctuations in the number of proteins produced per transcript. However, robustness against biochemical noise can be drastically enhanced by arranging the transcription factor binding domains on the DNA such that competing transcription factors mutually exclude each other on the DNA. We also elucidate the origin of the enhanced stability of the exclusive switch with respect to that of the general switch; while the kinetic prefactor is roughly the same for both switches, the "barrier" for flipping the switch is significantly higher for the exclusive switch than that for the general switch.