Hmm. Transportation. That's a little tougher. I'm not aware of any bus service to DV, but you could call Greyhound and see if they service any of the towns around there. You'd have to walk from there. You could also call the National Park Service in DV and see if they know of any public transit. Honestly, it may be tough without a rental car in DV.
The Santa Monica Mtns while perhaps not as spectacular as DV are far more accessible by bus. See
http://www.backpacker.com/february_2008_destinations_how_to_thruhike_la_county/destinations/12080?page=1 and
http://www.nps.gov/samo
I believe the Gabrieleno National Recreation Trail in the San Gabriel Mtns is fairly accessible by bus, but I'd have to check the
Metro schedules. I've run into people who have come to the San Gabriels by bus. There was also a
recent article in Backpacker Magazine about a trip to the San Gabriels by bus. You could also check the San Gabriels forum for further info:
http://sangabrielmnts.myfreeforum.org
While not a National Park, both of the above two locations offer some nice hiking and are close to Los Angeles which affords you far more transporation options.
Catalina island has public transportation available. I don't find Catalina to be the most scenic of places to hike (I like trees which are few on Catalina), but the coastline appeals to many.
I imagine the Santa Ana mountains in Orange County would be relatively accessible by public tranport, but inasmuch as I live in Los Angeles County, I won't pretend to know much about public transit in OC.
Things further afield are going to be difficult to reach without a car:
Death Valley National Park
Anza Borrego State Park
Los Padres National Forest
Joshua Tree National Park