Baby-IAXO is a new helioscope, a demonstration version of the proposed full-size IAXO, the International Axions Observatory. It is currently under design and once installed it will search via the Primakoff effect for axions or axion-like particles (ALPs) originating in the Sun. Axions can transform into photons in the presence of a significant magnetic field, and then be detected. The superconducting magnet for Baby-IAXO comprises two 10 m long racetrack coils, spaced by 0.8 m, operated in a quadrupole configuration and generating an average magnetic field of 2.1 T in the two 700 mm diameter free bores for axion-to-photons decay positioned in between the coils. Cool down and operational cryogenic requirements are handled by a combination of single and double stage cryocoolers. For the cool down of the 18.5 t cold mass within 22 days and for current leads cooling, the use of powerful single stage cryocoolers is of paramount importance. The cooling power is distributed across the cold mass and thermal shield using helium gas flow enforced by cryocirculators. The design of the 10 kA conduction cooled current leads is presented as well, since at 70 K these represent the dominant heat load. The cryogenic system allows for a constant heat load of 1.2 kW at the first stage and 8.5 W at the 4.2 K cold mass. Alternatively, the magnet may be operated in persistent mode, thereby reducing the heat load and the number of cryocoolers. It requires a more complex cooling circuit, implementation of a 10 kA persistent mode switch, and a delicate balance of energizing and operational heat loads.