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The development of a 5 T high temperature superconducting (HTS) insert magnet system is underway at the Korea Basic Science Institute (KBSI). The objective of the program is the design and fabrication of HTS insert magnet system for high field superconducting magnet applications. Since the HTS magnet system should be inserted into the conduction-cooled 15 T background magnet with 102 mm room-temperature bore, several ancillary requirements such as space restriction and cryostat structure need technical attention. The design and fabrication of the HTS insert cryostat are completed now. In this paper, the design of the cryostat for the 5 T HTS insert magnet is presented, taking into account the cryogenic loads and inventory of cryogenic liquid. In addition, the structure of the cryostat, minimizing thermal flow from room-temperature to cryogenic-temperature, is described in detail. The preliminary results of performance test of the HTS insert cryostat are also discussed.