Fourier-transform infrared (FTIR) magneto-spectroscopy is a powerful spectroscopic technique used to investigate many important effects in materials, e.g., electron spin resonance (ESR), cyclotron resonance, and transitions between Landau levels (LLs). Despite their enormous potential in material science, infrared (IR) magneto-spectrometers are still relatively rare and custom-made since such systems generally require complex infrastructure. This article presents a versatile broadband setup for FTIR magneto-spectroscopy spanning the range from THz/far-IR (FIR) to near-IR (NIR), high magnetic field up to 16 T, and cryogenic temperatures down to 2 K. It consists of a commercial FTIR spectrometer and 16 T cryogen-free superconducting magnet coupled with custom-designed optical coupling and transmission probes for experiments with various detectors. The functionality of the FTIR magneto-spectroscopic setup is demonstrated by the magneto-optical measurements on a cobalt-based single-molecule magnet (SMM) in the FIR region and germanium (Ge) in the NIR region.