You'd get a rainbow with brighter red and yellow, and dimmer green and blue.
What is often missed in discussion of "red stars" (like M class red dwarves) is that the surface temperature of those stars is similar to that of the tungsten filament in an old-style incandescent light bulb -- which, though fairly yellow when compared to a truly white light source, is still "white" enough that all colors are readily visible.
You can observe a full spectrum from a tungsten lamp -- using a CD as a diffraction grating, for instance -- just as you can from the Sun, or from a blue-white class O star. The difference is how bright the spectrum is in what color, but this isn't extremely obvious until the spectrum is pretty heavily skewed, either toward red or toward blue. A so-called "red dwarf" M star would still look white to the eye (just as a tungsten lamp filament does), so the spectrum it produces, while easily distinguished from that of a hotter star with instruments, will look very much the same to the human eye.
The width of the color bands is due to the physics of refraction working with human color perception. In truth, there are no "bands" of color in the rainbow -- it's a continuous, well, spectrum. We label certain parts (red, orange, yellow, green, blue, indigo, violet) based on how we perceive color, and those colors seem to form "bands" because we perceive them that way. The wavelengths of light, however, will still be evenly spread across the width of the rainbow's band, both the width and angular size of which are determined by the refractive index and dispesion of water. As long as the water is still water (and falling rain is pretty pure, in general), you'll get the same size, shape, and width rainbow we have as long as the light is visibly white to the eye (see above).