We examine the dynamics of antigenically diverse infectious agents using a mathematical model describing the transmission dynamics of arbitrary numbers of pathogen strains, interacting via cross-immunity, and in the presence of mutations generating new strains and stochastic extinctions of existing ones. Equilibrium dynamics fall into three classes depending on cross-immunity, transmissibility and host population size: systems where global extinction is likely, stable single-strain persistence, and multiple-strain persistence with stable diversity. Where multi-strain dynamics are stable, a diversity threshold region separates a low-prevalence, low-diversity region of parameter space from a high-diversity, high-prevalence region. The location of the threshold region is determined by the reproduction number of the pathogen and the intensity of cross-immunity, with the sharpness of the transition being determined by the manner in which immunity accrues with repeated infections. Host population size and cross-immunity are found to be the most decisive factors in determining pathogen diversity. While the model framework developed is simplified, we show that it can capture essential aspects of the complex evolutionary dynamics of pathogens such as influenza.