Galaxy clusters have long been theorized to quench the star formation of their members. This study uses integral-field unit observations from the K-band MultiObject Spectrograph (KMOS) - Cluster Lensing And Supernova survey with Hubble (CLASH) survey (K-CLASH) to search for evidence of quenching in massive galaxy clusters at redshifts 0.3<z<0.6. We first construct mass-matched samples of exclusively star-forming cluster and field galaxies, then investigate the spatial extent of their H{alpha} emission and study their interstellar medium conditions using emission line ratios. The average ratio of H{alpha} half-light radius to optical half-light radius (r_e,H{alpha}_/r_e,Rc_) for all galaxies is 1.14+/-0.06, showing that star formation is taking place throughout stellar discs at these redshifts. However, on average, cluster galaxies have a smaller r_e,H{alpha}_/r_e,Rc_ ratio than field galaxies: <r_e,H{alpha}_/r_e,Rc_> =0.96+/-0.09 compared to 1.22+/-0.08 (smaller at a 98 per cent credibility level). These values are uncorrected for the wavelength difference between H{alpha} emission and R_c_-band stellar light but implementing such a correction only reinforces our results. We also show that whilst the cluster and field samples follow indistinguishable mass-metallicity (MZ) relations, the residuals around the MZ relation of cluster members correlate with cluster-centric distance; galaxies residing closer to the cluster centre tend to have enhanced metallicities (significant at the 2.6{sigma} level). Finally, in contrast to previous studies, we find no significant differences in electron number density between the cluster and field galaxies. We use simple chemical evolution models to conclude that the effects of disc strangulation and ram-pressure stripping can quantitatively explain our observations.