We aim to identify and characterise binary systems containing red supergiant (RSG) stars in the Small Magellanic Cloud (SMC) using a newly available ultraviolet (UV) point source catalogue obtained using the Ultraviolet Imaging Telescope (UVIT) on board AstroSat. We select a sample of 560 SMC RSGs based on photometric and spectroscopic observations at optical wavelengths and cross-match this with the far-UV point source catalogue using the UVIT F172M filter, finding 88 matches down to m$_{F172M}$=20.3 ABmag, which we interpret as hot companions to the RSGs. Stellar parameters (luminosities, effective temperatures and masses) for both components in all 88 binary systems are determined and we find mass distributions in the ranges 6.1 to 22.3 Solar masses for RSGs and 3.7 to 15.6 Solar masses for their companions. The most massive RSG binary system in the SMC has a combined mass of 32 $\pm$4 M$_\odot$, with a mass ratio (q) of 0.92. By simulating observing biases, we find an intrinsic multipliciy fraction of 18.8 $\pm$ 1.5% for mass ratios in the range 0.3 < q < 1.0 and orbital periods approximately in the range 3 < log P [days] < 8. By comparing our results with those of a similar mass on the main-sequence, we determine the fraction of single stars to be ~20% and argue that the orbital period distribution declines rapidly beyond log P ~ 3.5. We study the mass-ratio distribution of RSG binary systems and find that a uniform distribution best describes the data below 14 M$_\odot$. Above 15 M$_\odot$, we find a lack of high mass-ratio systems.