The effects of variable cryogenic temperature on the tensile response are reported for the commercial superconducting composite wires consisting of niobium-titanium filaments in a copper matrix (NbTi/Cu). A variable temperature cryostat system is constructed to provide the successive cooling environment from room temperature (RT) to the liquid nitrogen temperature (LNT), and a cryogenic-type extensometer is used to measure tensile strains of the superconducting wires. With the variable cryogenic temperature, the corresponding stress and strain relations are recorded during the wire being stretched. The mechanical behavior of the NbTi/Cu composite wire and pure NbTi filaments, including the tensile strength and elongation at fracture, yield strength, and Young's modulus, are further captured. It is shown that ultimate tensile strengths and Young's moduli of the NbTi/Cu composite wire and pure NbTi filaments are almost linearly dependent on the temperature, whereas the elongation at fracture and the yield strength exhibit notable nonlinear features with the superconducting wires' cooling. In addition, based on the mechanical properties measured from RT to LNT, the properties at the lower temperature of 4.2 K are extrapolated for the NbTi/Cu composite wire and NbTi filaments, which are compared with the experimental data in the literature.