Context. The multiplicity properties of massive stars are one of the important outstanding issues in stellar evolution. Quantifying the binary statistics of all evolutionary phases is essential to paint a complete picture of how and when massive stars interact with their companions, and to determine the consequences of these interactions. Aims. We investigate the multiplicity of an almost complete census of red supergiant stars (RSGs) in NGC 330, a young massive cluster in the Small Magellanic Cloud. Methods. Using a combination of multi-epoch HARPS and MUSE spectroscopy, we estimate radial velocities and assess the kinematic and multiplicity properties of 15 RSGs in NGC 330. Results. Radial velocities are estimated to better than ±100 m s−1 for the HARPS data. The line-of-sight velocity dispersion for the cluster is estimated as σ1D = 3.20 +0.69−0.52 km s−1. When virial equilibrium is assumed, the dynamical mass of the cluster is log (Mdyn/M⊙) = 5.20 ± 0.17, in good agreement with previous upper limits. We detect significant radial velocity variability in our multi-epoch observations and distinguish between variations caused by atmospheric activity and those caused by binarity. The binary fraction of NGC 330 RSGs is estimated by comparisons with simulated observations of systems with a range of input binary fractions. In this way, we account for observational biases and estimate the intrinsic binary fraction for RSGs in NGC 330 as fRSG = 0.3 ± 0.1 for orbital periods in the range 2.3 < log P [days] < 4.3, with q > 0.1. Using the distribution of the luminosities of the RSG population, we estimate the age of NGC 330 to be 45 ± 5 Myr and estimate a red straggler fraction of 50%. Conclusions. We estimate the binary fraction of RSGs in NGC 330 and conclude that it appears to be lower than that of main-sequence massive stars, which is expected because interactions between an RSG and a companion are assumed to effectively strip the RSG envelope.