We present the results of near-infrared [Fe II] and H2 line imaging and spectroscopic observations of the supernova remnant 3C 396 using the Palomar 5 m Hale telescope. We detect long, filamentary [Fe II] emission delineating the inner edge of the radio emission in the western boundary of the remnant in imaging observations, together with a bright [Fe II] emission clump close to the remnant center. There appears to be faint, diffuse [Fe II] emission between the central clump and the western filamentary emission. The spectroscopic observations determine the expansion velocity of the central clump to be ~56 km/s. This is far smaller than the expansion velocity of 3C 396 obtained from X-ray observations, implying the inhomogeneity of the ambient medium. The electron number density of the [Fe II] emission gas is < 2,000 cm-3. The H2 line emission, on the other hand, lies slightly outside the filamentary [Fe II] emission in the western boundary, and forms a rather straight filament. We suggest that the [Fe II] emission represents dense clumps in the wind material from the red supergiant phase of a Type IIL/b progenitor of 3C 396 which have been swept up by the supernova remnant shocks. The H2 emission may represent either the boundary of a wind bubble produced during the main-sequence phase of the progenitor or molecular clumps left over inside the bubble. We propose that the near-infrared [Fe II] and H2 emission observed in several supernova remnants of Type IIL/b SNe likely has the same origin.