Short answer: the refraction of light is ultimately dependent on both temperature and exact composition of the medium (e.g., liquid water or ice) through which it passes. Note: strictly speaking, "steam" typically refers to water vapor at temperatures above the boiling point, but in casual usage (and for the purposes of this answer) it is commonly used to describe condensed water droplets occurring in such environments.
For background, rainbows, and refraction of light in general, occur as light passes between different media (in this case, air and liquid or frozen water) in which the speed of light differs. This property is described by the refractive index, the ratio of the speed of light in a vacuum to its speed passing through a particular medium. A generalized example of refraction as a ray of light passes between media with differing refractive indices is shown here:
More specifically, however, the amount of refraction depends on both the properties of the medium itself and on the frequency/wavelength of the light itself. For the example of a rainbow, the color pattern is produced by the slight variations in refraction over the range of visible wavelengths for liquid droplets occurring in generally similar environmental conditions:
Historically, the refractive index has been derived for a large number of media including air, ice, and both pure liquid water and various liquid water solutions at a variety of temperatures, all with respect to specific wavelengths of light. Thus, the differing refractive indices therein show that both the temperature and the composition (including any impurities) of liquid water or ice will indeed affect how light is refracted through them. Any more specific details related to the resulting effects would naturally depend on knowing the exact characteristics in question - one particular example, noted below by Keith McClary, is that the larger index of refraction for salt water causes a slightly smaller radius in the resulting rainbow.
Specifically for pure liquid water, while the characteristics of ice crystals and the effects of changing composition are likely too specific to estimate, water's refractive properties may be reasonably estimated over a range of temperatures. For example, Bashkatov and Genina (2002) present in their Table 1 examples of refractive indices for pure liquid water at temperatures between 0 and 100°C and for a range of wavelengths encompassing the visible light spectrum:
As shown, the differences in refractive index are relatively small even over this range of temperatures, and the relative differences over the visible wavelengths are similar for a given temperature, suggesting that a typical rainbow may well be observed in association with water droplets condensing from steam at high temperatures.