We visited Mesa Verde National Park in early July, 2011. The Anasazi really did not like company! Recommend visiting at least once even though it is really out of the way.
Plan on the visit taking the entire day. The park is about 1.5 hours west of Durango which is a great place to spend the night the day before launching your visit. While you are there you might as well spend another day and ride the train to Silverton