We report on the observation of gate-tunable proximity-induced superconductivity and multiple Andreev reflections (MARs) in a bulk-insulating BiSbTeSe2 topological insulator nanoribbon (TINR) Josephson junction with superconducting Nb contacts. We observe a gate-tunable critical current (IC) for gate voltages (Vg) above the charge neutrality point (VCNP), with IC as large as 430 nA. We also observe MAR peaks in the differential conductance (dI/dV) versus DC voltage (Vdc) across the junction corresponding to sub-harmonic peaks (at Vdc = Vn = 2ΔNb/en, where ΔNb is the superconducting gap of the Nb contacts and n is the sub-harmonic order). The sub-harmonic order, n, exhibits a Vg-dependence and reaches n = 13 for Vg = 40 V, indicating the high transparency of the Nb contacts to TINR. Our observations pave the way toward exploring the possibilities of using TINR in topologically protected devices that may host exotic physics such as Majorana fermions.