Carbon isotopic abundances (C-12 and C-13) are determined for 45 red giants in the CN-bimodal globular clusters M4 and NGC 6752 through low-resolution infrared spectroscopy of the first-overtone CO bands. All of the red giants are found to have quite low isotope ratios (C-12/C-13 approximately 3-10), indicating that all of these low-mass, low-metallicity giants have undergone extensive mixing of their outer envelopes with material processed through the CN cycle. It is also found that those giants with the strongest CN bands have the lowest C abundances: this implies an anticorrelation of the C and N abundances and again suggests the presence of various fractions of CN-processed material mixed with unprocessed material in the outer envelopes of these giants. In addition, low-resolution spectra of the violet CN (3883 angstrom) and CH (4300 angstrom) bands in main-sequence and subgiant stars in the globular clusters NGC 6752, M5, and 47 Tuc are found to exhibit CN and CH anticorrelations, implying that even main-sequence stars have differing (and anticorrelated) abundances of C and N. It is argued that, with all of the observations taken together, the CN-bimodal globular clusters most probably have "primordial" abundance variations of C and N, as evidenced by the main-sequence stars, while all of the low-mass giants undergo extensive mixing as they ascend the red giant branch, as evidenced by low C-12/C-13 ratios (a result in agreement with previous studies of field, open cluster, and globular cluster red giants): these "mixed" red giants will still exhibit evidence of their initial, main-sequence C and N abundance variations. It is speculated that an early generation of intermediate-mass asymptotic giant branch stars polluted the early globular cluster environment with slow stellar winds composed of CN-processed material.