We study dynamical capture binary black hole-neutron star (BH-NS) mergers focusing on the effects of the neutron star spin. These events may arise in dense stellar regions, such as globular clusters, where the majority of neutron stars are expected to be rapidly rotating. We initialize the BH-NS systems with positions and velocities corresponding to marginally unbound Newtonian orbits, and evolve them using general-relativistic hydrodynamical simulations. We find that even moderate spins can significantly increase the amount of mass in unbound material. In some of the more extreme cases, there can be up to a third of a solar mass in unbound matter. Similarly, large amounts of tidally stripped material can remain bound and eventually accrete onto the BH---as much as a tenth of a solar mass in some cases. These simulations demonstrate that it is important to treat neutron star spin in order to make reliable predictions of the gravitational wave and electromagnetic transient signals accompanying these sources.