  The first detection of gravitational waves from a neutron star - neutron star
(NS-NS) merger, GW170817, and the increasing number of observations of short
gamma-ray bursts (SGRBs) have greatly motivated studies of the origins of NS-NS
and neutron star - black hole (NS-BH) binaries. We calculate the merger rates
of NS-NS and NS-BH binaries from globular clusters (GCs) using realistic GC
simulations with the \texttt{CMC} cluster catalog. We use a large sample of
models with a range of initial numbers of stars, metallicities, virial radii
and galactocentric distances, representative of the present-day Milky Way GCs,
to quantify the inspiral times and volumetric merger rates as a function of
redshift, both inside and ejected from clusters. We find that over the complete
lifetime of most GCs, stellar BHs dominate the cluster cores and prevent the
mass segregation of NSs, thereby reducing the dynamical interaction rates of
NSs so that at most a few NS binary mergers are ever produced. We estimate the
merger rate in the local universe to be $\sim\rm{0.02\,Gpc^{-3}\,yr^{-1}}$ for
both NS-NS and NS-BH binaries, or a total of $\sim 0.04$~Gpc$^{-3}$~yr$^{-1}$
for both populations. These rates are about 5 orders of magnitude below the
current empirical merger rate from LIGO/Virgo. We conclude that dynamical
interactions in GCs do not play a significant role in enhancing the NS-NS and
NS-BH merger rates.
