Hi Dennis,
We've been to the Dolomites a number of times (3 trips in Oct. and 1 trip in late April/early May). No, everything will not be shut down. It will be quieter than say summer, and of course ski season, but enough will be open so you won't be too limited concerning lodging and restaurant choices.
While we love staying in Alta Badia, I'd suggest staying in any of the 3 towns/villages (Ortisei, St. Christina, Selva) of the Val Gardena. Ortisei is the largest and has a nice, although somewhat small, pedestrian zone. The Val Gardena offers more in the way of lodging, restaurants, shops, etc. than Alta Badia, especially in early May.
If you're traveling by car, I wouldn't let the fact that most (if not all) lifts are closed bother me. The Dolomite Passes near Val Gardena (Gardena and Sella Passes) are simply spectacular for up close, dramatic Dolomite scenery. You'll find some places for parking along the passes (we stopped in a parking area along the Sella Pass) with trails leading into the Dolomites. We saw quite a few hikers walking into the Dolomites from this one (along the Sella) in particular.
We stayed at and can recommend the Garni Ariston in St. Christina, Val Gardena.
www.garniariston.com
By the way, the 3 towns/villages that comprise the Val Gardena are just a few minutes drive from one to the other.
In Alta Badia, we love the Ciasa Montanara in La Villa.
www.montanara.it
If interested, we have some photo's at:
www.worldisround.com/home/pja1/index.html
www.worldisround.com/home/pja1/travel15.html
These websites are worth looking into:
www.val-gardena.com
www.altabadia.com
Hope this helps.
Paul