Deep JHK(s) photometry was obtained toward eight dense molecular cores, and J-H versus H-K-s color-color plots are presented. Our photometry, sensitive to the detection of a 1 M-., 1 x 10(6) yr old star through approximate to 35 - 50 mag of visual extinction, shows no indication of the presence of star/disk systems based on J-H versus H - K-s colors of detected objects. The stars detected toward the cores are generally spatially anticorrelated with core centers, suggesting a background origin, although we cannot preclude the possibility that some stars detected at H and K-s alone or at K-s alone are not low-mass stars or brown dwarfs (< 0.3 M-.) behind substantial amounts of visual extinction (e.g., 53 mag for L183B). Lower limits to optical extinctions are estimated for the detected background stars, with high extinctions being encountered, in the extreme case ranging up to at least A(V) = 46, and probably higher. The extinction data are used to estimate cloud masses and densities that are comparable to those determined from molecular-line studies. Variations in cloud extinctions are consistent with a systematic nature to cloud density distributions, and column density variations and extinctions are found to be consistent with submillimeter-wave continuum studies of similar regions. The results suggest that some cores have achieved significant column density contrasts (&SIM;30) on subcore scales (&SIM;0.05 pc) without having formed known stars.