Electrostatic tuning of Josephson junctions based on hybrid structures between a superconductor and a semiconductor or a topological material is essential to control supercurrents and the physics of these junctions. In this work, we investigate electric field-tunable Josephson junctions fabricated from a cadmium arsenide (Cd3As2) quantum well, which is a two-dimensional topological insulator. We show that the induced supercurrent can be completely suppressed by tuning the Fermi level into the charge neutrality gap of the quantum well. We observe a pronounced non-monotonic dependence of both the critical and excess currents on the carrier density. Both properties are maximized when the Fermi level is near the conduction subband edge. We explain this observation by considering the role of junction transparency in controlling supercurrents carried by Andreev bound states. We also detect features in differential conductance of the junctions that can be attributed to multiple Andreev reflections to high orders.