The critical behaviors of the antiferromagnetic spin-3/2 Ising model are studied by using the exact recursion relations (ERR) in the presence of a random crystal field on the Bethe lattice (BL). The sublattice order parameters, magnetizations, and quadrupolar moments are obtained in terms of ERRs under the effect of random crystal field which was either turned on or off with probability p and 1 - p, respectively, in a bimodal form. The nature of phase transitions, thus, the phase diagrams of the model are calculated on the (H, T) planes for the randomly changing crystal field values with probability p for the coordination numbers q = 3 and 4. It is found that the model exhibits both second- and first-order phase transitions in addition to some critical points, i.e., tricritical point (TCP), critical end point, double critical point, and isolated end point. We have also observed the reentrant behavior for some values of our system parameters.