Two-dimensional (2D) Janus monolayers with mirror asymmetry have been found to possess extraordinary physical characteristics, meaning they could find applications in optoelectronic, electronic, and electromechanical devices. However, 2D Janus materials with intrinsic magnetism are rare. The electronic structure and magnetic properties of Janus Cr2I3X3 (X = Br, Cl) monolayers are investigated by first-principles calculations. Janus Cr2I3X3 monolayers are an indirect band gap semiconductor with good stability, intrinsic ferromagnetism, and electric polarization. Cr2I3Br3 monolayer is a half semiconductor with perpendicular magnetic anisotropy (PMA), while Janus Cr2I3Cl3 monolayer is a bipolar magnetic semiconductor with in-plane magnetic anisotropy (IMA). The band gap and Cr magnetic moment of Janus Cr2I3X3 monolayers can be tailored by biaxial strain. Additionally, a strain-induced transition from half semiconductor to bipolar magnetic semiconductor, and reversal between PMA and IMA, appears in the Janus Cr2I3X3 monolayers. These results enrich the diversity of Janus 2D materials, which have potential applications in 2D spintronic devices.