What would be the best neighborhood(s) in Denver for easy access to the airport?

I vote for Park Hill. Great historic neighborhood, close to the zoo, the Museum, City Park. It's tough to beat if you want to be close to the airport.

Definitely check out Green Valley Ranch or Reunion (which technically has a Commerce City address but is nothing like it). Green Valley Ranch has less expensive homes but is literally minutes from DIA and many people who work there live in Green Valley Ranch. Reunion is higher priced but in my opinion much nicer. The homes are more expensive but it is a very pretty area with a nice lake and huge rec center. Just from my experience there is a lot of traffic on Tower Road at peak times and it is way past time for expanding the road, so either way you're going whether it's GVR or Reunion it's usually best to travel via the tollway.

If you want to actually be in Denver and relatively close to what's going on in town as well, I'd recommend checking out Stapleton. It used to BE the airport, and it's probably about a 15-20 minute drive with no traffic to the "new" airport, which isn't bad compared to airport access for the rest of the city. And it's a fun, mixed-use neighborhood with restaurants, coffee shops, etc.