Cluster galaxies are affected by the surrounding environment, which influences, in particular, their gas, stellar content, and morphology. In particular, the ram pressure exerted by the intracluster medium promotes the formation of multiphase tails of stripped gas detectable both at optical wavelengths and in the submillimeter and radio regimes, tracing the cold molecular and atomic gas components, respectively. In this work we analyze a sample of 16 galaxies belonging to clusters at redshift ∼0.05 showing evidence of an asymmetric H i morphology (based on MeerKAT observations) with and without a star-forming tail. To this sample we add three galaxies with evidence of a star-forming tail and no H i detection. Here we present the galaxies’ H2 gas content from APEX observations of the CO (2–1) emission. We find that in most galaxies with a star-forming tail the H2 global content is enhanced with respect to undisturbed field galaxies with similar stellar masses, suggesting an evolutionary path driven by the ram pressure stripping. As galaxies enter into the clusters, their H i is displaced but also partially converted into H2, so that they are H2 enriched when they pass close to the pericenter, that is, when they also develop the star-forming tails that are well visible in UV or B broad bands and in Hα emission. An inspection of the phase-space diagram for our sample suggests an anticorrelation between the H i and H2 gas phases as galaxies fall into the cluster potential. This peculiar behavior is a key signature of the ram pressure stripping in action.