I read the fables as a child and surprisingly I feel it contributed to shaping my perception of moral values similar to parables of the Bible which I was not exposed to as a child. Which ever format of teaching moral life lessons to your child you choose, be it religious or fables I am in agreement that it is a good idea.
Aseops fables and biblical parables have shaped are vocabulary of describing human behaviour by comparing it to certain animals. If someone is called a snake or a fox we would make the assumption that this person is evil or sly. This is because these animals have been used in famous stories and be demonised to a degree, like the snake who tempted Eve or the cunning fox in "The fox and the crow".
As a teacher of English language I used the Aesop's story: The Town Mouse and The Country Mouse, in teaching English language as a second language to pre-schoolers (aged 4-6). It was very useful and fun for kids. I had my own version for the story (there are many versions of his stories although the originals are better). I have two hubs on this topic.