STIS white-light coronagraphic imaging has been carried out for 14 nearby, lightly reddened Herbig Ae stars, providing data on the environments and disks associated with these stars. No disks are detected in our data when the Herbig Ae star is accompanied by a stellar companion at r ≤ 2''. We find that the optical visibility of protoplanetary disks associated with Herbig Ae stars at r ≥ 50-70 AU from the star is correlated with the strength of the mid-IR PAH features, particularly 6.2 μm. These features, like the FUV fluorescent H2 emission, trace the presence of material sufficiently far above the disk midplane that it is directly illuminated by the star's FUV radiation. In contrast, measures of the bulk properties of the disk, including ongoing accretion activity, mass, and the submillimeter slope of the SED, do not correlate with the surface brightness of the optical nebulosity. Modelers have interpreted the appearance of the IR SED and the presence of emission from warm silicate grains at 10 μm as a measure of geometrical shadowing by material in the disk near the dust sublimation radius of 0.5 AU. Geometrical shadowing sufficient to render a disk dark to distances as large as 500 AU from a star would require that the star be optically visible only if viewed essentially pole-on, in disagreement with our program star system inclinations. Rather than invoking shadowing to account for the optically dark disks, the correlation of the STIS detections with PAH emission features suggests a correlation with disk flaring and an anticorrelation with the degree of dust settling toward the midplane. If this correlation continues to lower levels, the STIS data suggest that improvements in coronagraph performance that suppress the residual scattered and diffracted stellar light by an additional factor of ≥10 should render the majority of disks associated with nearby Herbig Ae stars detectable.