I Suggest Age 9+
I think it was six or seven when I first watched this movie, and while I loved it, some scenes did scare me. (Warning - the below contains spoilers for the movie).
When Bastion (the boy) first begins reading the story, and it starts describing the forest, the loud noise made me jump because it was unexpected (even now I still jump). The Nothing can also be pretty scary, with it's sharp teeth and attempts to kill several characters.
Going to sleep, I didn't have any nightmares (that I recall) after first watching this movie, though I was always a little scared of the dark after watching it (thanks to the Nothing). Please note, however, that I was only scared of the dark for a short time after watching the movie.
I suggest age 9+ for watching The Never Ending Story alone, and 8 with parents, but then, all kids are different. Some 9 year olds might be scared of the movie, some may not.