Observations from NASA's Cassini spacecraft established that Saturn's moon Enceladus has an internal liquid ocean. Analysis of a plume of ocean material ejected into space suggests that alkaline hydrothermal vents are present on Enceladus's seafloor. On Earth, such deep-sea vents harbour microbial ecosystems rich in methanogenic archaea. Here we use a Bayesian statistical approach to quantify the probability that methanogenesis (biotic methane production) might explain the escape rates of molecular hydrogen and methane in Enceladus's plume, as measured by Cassini instruments. We find that the observed escape rates (1) cannot be explained solely by the abiotic alteration of the rocky core by serpentinization; (2) are compatible with the hypothesis of habitable conditions for methanogens; and (3) score the highest likelihood under the hypothesis of methanogenesis, assuming that the probability of life emerging is high enough. If the probability of life emerging on Enceladus is low, the Cassini measurements are consistent with habitable yet uninhabited hydrothermal vents and point to unknown sources of methane (for example, primordial methane) awaiting discovery by future missions.