Fish don't blink or close their eyes at all, even when they sleep, because they don't have eyelids. The only exception is the shark which has a nictitating membrane, which is a transparent type of extra eyelid that reptiles, birds and some fish have.
Scientists are still studying how fish sleep, since fish don't sleep in the same way that people do. Fish appear to have periods of rest, where they are in a trance-like state, floating in one spot or resting in a protected area. By reducing their activity, fish slow down their metabolism and conserve energy. Scientists suggest that this provides the same restorative rest for fish that people get after sleeping.