Children's fears tend to be associated with their age. Not all children have the same fear at the same age (or even the same fear); but in general, there's a set of "types of fears" that tends to be associated with each stage of development.
None of my kids had any fears that were much different from "the usual" for their age. I did nothing to help them work out the fears. I just respected that they were uncomfortable about one thing or another and waited the year or so until they outgrew the stage fairly naturally. If I got the the signal that it appeared the fear was beginning to ease up some, I would then do a little something to encourage my child's having a chance to dabble in being around something (like an animal) while he knew I was nearby and would make sure the animal couldn't do something that scared him.
I'd recommend parents of young children do some reading up on the ages at which the different types of fears tend to bother children, because - really - children do outgrow those fears (as long, I think, as someone doesn't try to force them to overcome the fear when they're not at the age when they're ready, and about to outgrow it anyway). My personal approach/opinion (not just with children, but with pets too) is that if they know they can count on your to make them feel safe and sure that you won't force them to have to deal with whatever it is they're afraid of, they learn to trust you so much that when the times comes when they're more ready to move past their fear, it's almost effortless to support their doing that. When they know you're around to protect them from what they fear (or sometimes from letting what they fear get too close to them), they feel sure and comfortable enough, and trust you enough, that they're more willing to venture a little more into being around what they fear when they're a little more ready and/or have had a little more limited/supervised exposure to something.