The adsorption of surfactant molecules at the droplet-vapor interface can affect the droplet surface tension and thus alter the rate of surface-stimulated crystal nucleation. Recently, we have shown that if the condition of partial wetting holds for at least one crystal facet then the crystallization process is most likely to initiate at the droplet surface. Here, we outline an adsorption model to determine the main physical characteristics of the interior and surface layer of a binary droplet. The theoretical model is illustrated by numerical calculations for aqueous nitric acid droplets. As expected, an increase in the surfactant concentration in the droplet interior lowers the droplet surface tension, which has an impeding effect on the surface-stimulated crystallization. However, we show that an increase in the surface-to-volume ratio of a droplet favors crystal nucleation at the surface not only kinetically but also thermodynamically. This occurs because for a given droplet composition smaller droplets have a higher surface tension. Thus, the impeding effect of a soluble surfactant on the surface mode of crystal nucleation becomes weaker as the droplet size decreases. Our results also show that for a fixed overall droplet composition the excess surface coverage decreases with decreasing droplet radius. The effect of organic contaminants on crystallization in aqueous nitric acid droplets is also qualitatively studied.