You definitely won't see Mykonos in its best light in August.
If you want to avoid crowds, avoid the town. It will be packed and I don't imagine it would be the least bit enjoyable to walk around, especially with your two young children.
Ornos will be very busy, too, especially since it's one of the top family beaches on the island. You could take the bus to get to/from Town, but the buses will be very busy also. Taxis are a possibility, but with only three dozen cabs on the entire island, and August being peak season, they are going to be in high demand and you could face a significant wait to catch a taxi in either direction.
When you're checking for hotels, whether in Town or at a beach, don't be surprised if you have trouble finding availability -- in August, many require a minimum stay of 3 nights or longer.
One option you could consider is Makis Place near the New Port at Tourlos. They have a swimming pool plus a children's pool, and offer transportation to and from the port. Agios Stefanos beach, another popular family beach on Mykonos, is less than a 20-minute walk away (and along the bus route between the beach and Mykonos Town). The Town itself is walking distance; however, it's a long a narrow road with no sidewalks, and in August this road will be extremely busy with traffic -- the road won't be pleasant to walk. But you wouldn't have to go into Town at all if you didn't want to -- the Molaraki restaurant is right on the premises, and the excellent Matthew Taverna is practically next door.
Hotel website: http://makisplace.com/
Edited: 3:10 pm, February 03, 2013