Background Superconducting undulator (SCU) prototype with small magnet gap of 5 mm, long magnet length of 4 m and high magnet field of 1.58 T was being developed at Shanghai High Repetition rate XFEL and Extreme light facility (SHINE). Compared to any other superconducting undulator, there is no cryocooler being installed on the cryostat in this SCU prototype. Purpose This study aims at the cooling design for the binary current leads for SCU's normal operating. Methods Binary current leads composed of normal conductive copper leads and high temperature superconducting current leads (HTS) were adopted for SCU to connect superconducting coils inside the cryostat and outer cables. Low-temperature helium gas was used to transport independent refrigerator system to the cooling tubes inside the prototype, hence the binary current leads were cooled. Thermal conduction components installed on the middle of the thermal shield were employed to transfer heat load of normal conductive copper leads, and heat load of copper leads was optimized by simulation. Auxiliary superconducting rods were designed for connecting cold ends of HTS in the cryostat test. Results The temperature difference between hot ends of HTS and low-temperature helium gas is less than 20 K from the result of cryostat test, all binary current leads is operating normally with full current. Conclusions It is practicable to use cooling tubes with low-temperature helium gas to cool binary current leads of the SCU prototype by thermal conduction, which is different from cooling solution for current leads in any other SCU being developed presently.