My brother and I went about 3 years ago and actually stayed in each of the towns. The first night we stayed in Cortez, found a decent Mexican restaurant for dinner, then got up early to visit Mesa Verde the next day. We only stayed in Cortez because we wanted the full day at the park. That evening, we stayed in Durango, which had a whole lot more in terms of amenities -- restaurants, bars, shops, microbrewery, etc. Walk-able downtown area. If you don't mind an hour drive to each of your activities, then I'd say stay in Durango. But we were very happy we stayed in Cortez the first night, and didn't have to get up too early.