Context. Galaxy clusters are among the largest and densest structures in the Universe. Their high density generally increases the suppression of star formation, known as quenching, altering galaxy properties. Aims. We study the quenching of emission-line galaxies (ELGs) in the rich cluster ZwCl 0024.0+1652 (Cl0024) at redshift z ∼ 0.4, aiming to determine if and how star formation is suppressed. Methods. Using multi-object spectroscopy from the GLACE survey, we extracted fluxes and redshifts of [O II]λλ3727, 3729, Hβ, and [O III]λ5007 emission lines to derive star formation rates (SFRs) for 173 ELGs. We also performed spectral energy distribution fitting to obtain key evolutionary parameters such as stellar masses (M⋆) and the 4000 Å break (D4000) index. Results. We derived the M⋆ ‑ SFR relation for 98 star-forming galaxies (SFGs), finding 34.7% exhibit suppressed SFRs in the cluster, compared to 11.0% in the field. While the SFRs show no significant variation with local density, the fraction of SFGs is 1.55 times higher in the cluster outskirts than in intermediate-density regions. The specific SFR decreases strongly with D4000 for active SFGs but remains constant for suppressed galaxies. The fraction of suppressed galaxies in the infall region is 2.6 times higher than in the core, especially in the infalling structure B of the cluster. The cluster's total mass does not appear to be a key factor in SFG quenching. Conclusions. Star formation in Cl0024 galaxies is suppressed by the dense cluster environment. This suppression is evident in SFG fractions and parameters tracing long-term evolution, indicating prolonged quenching. The SFGs preferentially reside in low-density regions, while suppressed galaxies dominate the infall region, supporting a 'delayed-then-rapid' quenching scenario.