Neutron monitors (NM) are ground-based cosmic ray detectors that measure the flux of primary cosmic rays at the GeV-energy range by counting (primarily) secondary neutrons in atmospheric cascades. The South Pole NM started in 1964 as an International Geophysical Year (IGY) monitor that operated until 1974. In 1977 three NM64 with boron-trifluoride proportional counters were installed. In January 2004, they were replaced with helium-3 detectors. Apart from a four-year gap (2005-2009), the NM has operated continuously since then. Over decades of its operation, a decline of the South Pole NM counting rate was observed, significantly larger than temporal changes observed by other NM stations. To investigate this decline, we simulated the counting rates of the NM64, for boron-trifluoride and helium-3 configurations, using the FLUKA Monte-Carlo package. We compare the observed count rates from 1964 to December 2021 with simulated count rates obtained using monthly values of the modulation parameter ($\phi$, in MV) determined in two different ways. The extended data set and the simulations indicate that most of the observed decline can be associated with the transition from the IGY to the NM64. Further work with a specific IGY simulation is required.