Not sure I ever saw a crowd at Delicate arch in the early morning. However, only did the early morning shoot twice in over six years. If you are going to do Delicate arch; early morning is the best for the walk (especially if there in the summer months), and the best time for few people. However, very early is when the lighting can get strange. Evening is the most popular due to the view of the La Sal's through the arch. However, evening, even in the hot summer months, can have hundreds of people and many with kids running in the bowl in front of the arch. However, when the sun starts going down, the serious photographers tell everyone to get the (you know what) out of the way.
For two early morning shots that are more famous than Delicate for early morning one is Mesa arch. One of the most photographed arches and this one is actually in Canyonlands and not Arches. Easy half mile, flat walk, can get rather crowded, but since I lived in Moab for over six years, I had time to pick the best day...
NIKON F5 - FUJI VELVIA OR PROVIA SLIDE FILM
Glow on the bottom of the arch is from the rising sun. And that close formation through the arch is known as washer woman and the monster. The back part if you look carefully, looks like a woman washing cloths, the front is the monster. (Hey, I didn't name them.)
The other famous sunrise is Turret Arch through windows arch. To get the perfect location, you need to climb a wall to get to a perch that will contain about three photographers with their equipment. Again, nice to live there, so that early 2AM climb is fun...
NIKON F5 - FUJI VELVIA OR PROVIA SLIDE FILM
And last...probably one of the best and also famous arch is not near Arches, but years ago a small plane flew through it. Decent 1-2 miles walk, a climb up a steep ledge can be hard with a lot of equipment, but there is a permanent rope and some foot/hand holds. But once up, a great place to have lunch and sit around and solve the problems of the world. Corona Arch
NIKON F5 - FUJI VELVIA OR PROVIA SLIDE FILM