/ AbstractWhile neutrinos are often treated as a background for many dark matter experiments, these particles offer a new avenue for physics: the detection of core-collapse supernovae. Supernovae are extremely energetic, violent and complex events that mark the death of massive stars. During their collapse stars emit a large number of neutrinos in a short burst. These neutrinos carry 99\% of the emitted energy which makes their detection fundamental in understanding supernovae. This paper illustrates how COSINUS (Cryogenic Observatory for SIgnatures seen in Next-generation Underground Searches), a sodium iodide (NaI) based dark matter search, will be sensitive to the next galactic core-collapse supernova. The experiment is composed of two separate detectors which will be sensitive to far and nearby supernovae. The inner core of the experiment will consist of NaI crystals operating as scintillating calorimeters, mainly sensitive to the Coherent Elastic Scattering of Neutrinos (CE$ν$NS) against the Na and I nuclei. The low mass of the cryogenic detectors gives the experiment a sensitivity to close supernovae below 1kpc without pileup. They will see up to hundreds of CE$ν$NS events from a supernova happening at 200pc. The crystals reside at the center of a cylindrical 230T water tank, instrumented with 30 photomultipliers. This tank acts as a passive and active shield able to detect the Cherenkov radiation induced by impinging charged particles from ambient and cosmogenic radioactivity. A supernova near the Milky Way Center (10kpc) will be easily detected inducing $\sim$60 measurable events, and the water tank will have a 3$σ$ sensitivity to supernovae up to 22kpc, seeing $\sim$10 events. This paper shows how, even without dedicated optimization, modern dark matter experiments will also play their part in the multi-messenger effort to detect the next galactic core-collapse supernova.