A cryogen recondensing cooling system was designed for a 7 T superconducting solenoid magnet in an electron beam ion source (EBIS) system. In an EBIS system, it is difficult to access the apparatus due to strong X-ray emission when in full operation. The proposed cooling system was adapted for liquid-helium recondensation. An EBIS is a typical ion source that is feasible for highly charged ion production. Thus, the EBIS system can meet various requirements of end applications, and the development of such a system is strongly required. Estimations of the thermal and structural stability levels are very important during the design of a cryostat; therefore, these values were calculated by means of a numerical analysis. In the thermal analysis, the heat loads were calculated according to the temperature gradient after calculating the temperature distribution. The cooling margins were estimated at the first and second stages of the cryocooler, respectively, with respect to the calculated heat load. It was also estimated that the thermal stress due to the temperature gradient. Finally, it was confirmed that the proposed cryostat design was thermally and structurally stable and suitable for application to a superconducting magnet for an EBIS system.