Published Resources

The Arabian Nights were translated from a variety of source materials written in Arabic or French. As can be seen, there is a massive amount of material. Some of the "best known" tales were such as Sinbad and Aladdin were actually first introduced in the later French language collections. The Translations below have sometimes paraphrased or reduced the content. See Irwin's Companion or Haddawy's book for detained discussions of the original sources.
There is much erotic material found in The Arabian Nights - they were originally stories for adults. Because of this, in considering the various translations of the Tales of the 1,001 Nights - there needs to be a sensitivity to their handling of Adult and Racial Content. The italicized comments below are taken from a Review by Ana Mardoll, who compares the passage in the opening Tale of King Shahryar and of his Brother, King Shahzaman where the latter observes the infidelity of the former's wife in the garden in her evaluation of several translations. That might be a place to look in comparing others.
There are quite a few "classic" children's fairy tales books drawing from the 1,001 nights. Some are listed below - e-book versions may be available for those in public domain, and Kindle books are available for many more recent editions.

• Arabian Nights. A Companion By Robert Irwin traces the development of the stories from prehistoric India and Pharaonic Egypt to modern times. It explores the history of the translation, and explains the ways in which its contents have been added to, plagiarized and imitated. Above all, the book uses the stories as a guide to the social history and the counterculture of the medieval Near East and the world of the storyteller, the snake charmer, the burglar, the sorcerer, the drug addict, the treasure hunter and the adulterer.
  
  
• The Arabian Nights and Sindbad: And other Stories from the Arabian Nights by Husain Haddawy (Translator) and Muhsin Mahdi (Editor). (2008) These translations are consider very faithful to the Arabic.

• The Arabian Nights (Barnes & Noble Classics) by Muhsin J. Al-Musawi 736 Pages - For children, however, the superior volume is probably the Muhsin al-Musawi edition. This edition is condensed, but the editing was done with great care to maintain story structure and content. The adult content has been toned down considerably, the racial content has been handled tactfully, the extra songs and poems have been almost entirely removed, and there are interesting and attractive pictures in the electronic edition.

• Penguin Classics Editions of Arabian Nights - Volume One (992 Pages) Volume Two {800 Pages} Volume Three (855 Pages) by Anonymous (Author),Robert Irwin (Editor, Introduction), Malcolm C. Lyons and Ursula Lyons (Translators) Aiming to be a complete version, these were translated from the French version by Galland. Rather than labelling the stories, the volumes refer simply to numbered "nights" - you have to go to the indexes in the back of each volume to find stories by their titles.

• The Book of the Thousand and one Nights. Volume One {656 Pages} Volume Two (608 Pages) Volume Three (576 Pages) Volume Four (544 Pages) by J.C Mardrus (Editor),E.P Mathers (Editor) 1991 - They've made a concerted effort to retain the adult content without being lewd, the racial content without descending into offensive caricature, the poetic content without overwhelming the reader, and the entire content without condensing the text and losing material.

• One Thousand and One Nights: A Retelling by Hanan al-Shaykh. Definitely not for children. A strongly feminist retelling of portions of the Tales.

• The Arabian Nights: Tales from a Thousand and One Nights (Modern Library Classics) Reissue Edition by Richard Burton (Translator), A. S. Byatt (Introduction) 1049 Pages -There are many published volumes and abridgements of Burton's work, originally six volumes. Done in the 1880's. Burton's translation has much offensive language,and much lewd sexual and racial content.

• The Arabian Nights, Volume I: The Marvels and Wonders of The Thousand and One Nights (Signet Classics) (592 Pages) and The Arabian Nights, Volume II: More Marvels and Wonders of the Thousand and One Nights 2010 by Jack Zipes (Adapter) (448 Pages) - These are revisions of Burton's Work. More readable, the language is improved, but the sexual and racial content is still very much there.

  Children's Versions

• One Thousand and One Arabian Nights (Oxford Story Collections) by Geraldine McCaughrean - A delightful telling of some of the stories.

• The Arabian Nights by Andrew Lang 1898 Thin on content and heavily paraphrased. The stories are gutted to remove the adult content and shorten the tale length for children, but in many cases the changes are not carefully glossed over, and huge plot holes and unresolved threads are left dangling.
  

• The Arabian Nights by Anna Tweed 1910 - Illustrated by Caspar Emerson and Leon D'Emo

• Stories from the Arabian Nights (192 Pages) and Sinbad the Sailor and Other Stories from the Arabian Nights by Laurence Housman (Author) 1911 - Illustrated by Edmund Dulac

• Fairy Tales From The Arabian Nights by E. Dixon (Author) Illustrated by John D. Batten