You should decide on how to spend your time in the Dolomites, do you want to drive mountain passes, hike, walk around a town and take lifts for easy walks, etc... You only have limited time so cannot do everything and should choose a town/area.
In addition to the lifts on the passes which are listed and intended more to be visited if driving a car through the passes thus taking up a full day.
From the town of Ortisei which people have recommended there are 3 lifts to 3 different areas which is why many recommend staying there. You can leave your car parked at your hotel and easily reach 3 different elevated areas (Alpe Di Suisi, Seceda and Rasciesa)
All 3 of those lifts are open all month.
No other town has 3 lifts in town. Nearby Santa Christina is very close and has a couple of lifts to the same areas I believe.
Others want to stay somewhere with a great view, in that case there are very nice resorts on top of Alpe Di Suisi, cannot imagine a better place for a view other than mountain top rifugios in the area but usually if staying In/on Alpe Di Suisi you probably would not leave that area for your short amount of time since visiting other areas from there would be inconvenient.
Castelrotto is close to Alpe Di Suisi and probably has a lift up there but is the only area it has a lift to and is not on Alpe Di Suisi. The town is smaller than Ortisei so less to do so for most I think either staying in the towns of Ortisei/Santa Christina or on the Alpe Di Suisi would be preferred to Castelrotto depending on what your interests are.
In the town of Cortina the lifts which is on the east side and definitely closer to Venice (further from Verona) that are there all close sometime in early Sept so if intending to stay there you would want to check the lift schedule/dates closely.
This is a ritzy town but I thought most places there look a little dated, it is closer to more World War I historical areas and much closer to the famous Tre Cime which is national park with great hiking and a toll road. There are nice lakes near Cortina as well.
If you are really into nature and hiking, a rifugio stay would be really unique and more memorable than a hotel but at this late date I doubt you could get a private room so envision a bunk style hostel type sleeping arrangement, shared bath maybe without hot showers, etc... very rustic.
if I were traveling solo staying in the Rifugio Locatelli right at the base of the Tre Cime a decent hike from civilization would be a dream stay for me but not happening with wife and daughter with me.
I would recommend picking one town/area to stay your time in not break it up.