We present a study of the long-term evolution of coronal mass ejections (CMEs) observed by the Large Angle and Spectrometric Coronograph (LASCO) on board SOHO during the ascending, maximum, and part of the descending phases of solar cycle 23 and their relation with the modulation of galactic cosmic-ray (GCR) intensity observed at 1 AU by the Climax neutron monitor and IMP-8 spacecraft. We compare the long-term GCR modulation with the CME occurrence rate at all, low, and high latitudes, as well as the observed CME parameters (width and speed). Twenty-seven day averages of CME occurrence rates and CME properties from 1996 January to 2003 December are presented in the Appendix. The general anticorrelation between GCR intensity and the CME rate is relatively high (~-0.88). However, when we divide the CME rate into low- and high-latitude rates and compare them with the GCR intensity during the ascending phase of solar cycle 23, we find a lower anticorrelation between the low-latitude the CME rate and GCR intensity (~-0.71) and a very high anticorrelation between the high-latitude CME rate and GCR intensity (~-0.94). This suggests that, in general, CMEs could cause the decrease in the GCR flux in the inner heliosphere, as stated by the global merged interaction region (GMIR) theory. In particular, during the ascending phase of cycle 23 (qA > 0), this flux comes mainly from heliospheric polar regions. Thus, high-latitude CMEs may play a central role in the long-term cosmic-ray modulation during this phase of the cycle by blocking the polar entrance of GCRs to the inner heliosphere. This study supports the scenario in which CMEs, among other structures, are the building blocks of GMIRs, although we propose that the spherical shells (GMIRs) are closed separately at polar and equatorial regions by CMEs of different latitudes. Our results suggest that all CME properties show some correlation with the GCR intensity, although there is no specific property (width, speed, or a proxy of energy) that definitely has a higher correlation with GCR intensity.