The production of radioactive isotopes by interactions of cosmic-ray particles with sodium iodide (NaI) crystals can produce radioactive backgrounds in detectors used to search for rare events. Through controlled irradiation of NaI crystals with a neutron beam that matches the cosmic-ray neutron spectrum, followed by direct counting and fitting the resulting spectrum across a broad range of energies, we determined the integrated production rate of several long-lived radioisotopes. The measurements were then extrapolated to determine the sea-level cosmogenic neutron activation rate, including the first experimental determination of the tritium production rate: $(80\ifmmode\pm\else\textpm\fi{}21)\text{ }\text{ }\mathrm{atoms}/\mathrm{kg}/\mathrm{day}$. These results will help constrain background estimates and determine the maximum time that NaI-based detectors can remain unshielded above ground before cosmogenic backgrounds impact the sensitivity of next-generation experiments.