This is a very interesting subject and one I have spent a great deal of research on it over the years.
My research on the subject has been more about the moon and the behaviour of whitetail deer, but since all animals including humans are supposed to be under the influence of the moon this will also go for fish.
Depending to who you listen, the moon has a large influence over animals. However, I have found that the most avid moon phase advocates invariably also have a products for sale or make otherwise a living of their moon knowledge.
My own research, taking field notes from observations, shows clearly that their is not a big difference in the movement of animals during any given moon phase. Having said that, animals seem to move a little more during a full moon phase. This has more to do with practicality than with the moon itself, because on overcast nights the deer would not move more on on full moon days where the sky was open. During a full moon there is more light, therefore animals can see better and thus move more since they have not to stay in the cover for fear of not being able to see predators in the dark.
Many hunters and fishers believe in the moon phases and it is this believe that gives them more energy, more will and lets them stay longer in the woods, or in this case on the water. Next to knowledge hunting or fishing has a lot to do with believing in something, be that the moon, a bait or other the tackle we use this fact alone has a big influence in our hunting/fishing success.
What I did find out without a doubt is that animals are under the weather. This means they move and feed more heavily before a severe weather front moves trough an area and right after that weather front passes. I also find that animals react and adjust their movement to hunting/fishing pressure. On weekends when there are many fishers on a particular body of water the fish might not bite. Yet go to the same place on a Monday and you might have the best time of your life, simply because the fish "know" Sundays are bad for them and Mondays are safe. The same is true when the weather is not very nice, since most fishers are "fair weather" fishers.