Your question is a really interesting one. I went to work last Monday (I work for the National Park Service as a park ranger) and asked your question to several of my co-workers. I think we're all mulling on it a bit, so we may come up with other ideas as we ponder our personal experiences within the National Park System.
Here are some ideas that I came up with:
There are certainly many urban historical national parks that are near major cities, which could be accessed by air or Amtrak and from there by either bus or, lacking that, by taxi that could drop you off. Parks that jump to mind would be Cuyahoga Valley National Park and Valley Forge, Great Falls of the Potomac, the Mall in Washington, D.C., etc. I'd guess that many of the national parks around San Francisco would be accessible by mass transit as well. Petroglyphs is on the edge of Alburqurque and is certainly accessible by someone without a car. You could fly into Jackson Hole to visit Grand Teton National Park.
For something more adventurous, I'd suggest the Dry Tortugas National Park. Dry Tortugas (Florida) could be accessed by flying into Key West and then taking one of the two ferries to the Tortugas islands. The Dry Tortugas were by far and away our favorite spot to visit on a trip to Florida a couple of years ago. We camped on the island, which is the best way to see the park in my estimation because as the ferries pull out in the afternoon you're pretty much alone. (Great skies, marvelous history and architecture, lots of wildlife, nice beaches, coral reefs...)
For Denali, one could take Amtrak or fly to Seattle and catch the Alaskan ferry system up the Inside Passage to Whittier, AK. Hop the Alaskan Railroad north to Denali, then take the park's bus system into the interior of the park. Check out both the Alaskan Maritime and Alaska Railroad websites for all sorts of options. This is one of my dream trips.... Another Alaska trip would be to fly into Fairbanks; there is a tour company that runs up the Haul Road (Dalton Highway) through the magnificent Brooks Range, out onto the North Slope and all the way to Deadhorse. I've been on the Haul Road and it was fantastic--lots of scenic and wildlife value.
One other option is to go on an organized tours. By doing so, you remove much of the issue of how to get to a national park or other area of great scenic value. For instance, numerous tour companies run tours to the Canadian Rockies or loops in the Pacific Northwest, etc. They may be a bit pricey.
A less expensive way may be to check out Exploritas (formerly Elderhostel, which has dropped its age requirement). I believe Exploritas starts many of their trips from major transportation hubs, even though some of the trips may go into pretty wild places.
One issue with organized tours is that they may not offer much time for a serious photographer, however, some organizations offer tours that cater to photographers (such as Exploritas). However, if you enjoy hiking or canoeing be sure to check out Wilderness Inquiry. They run tours all over North America and other places as well. Many of their trips enter wilderness areas at a relatively low price and they also pick up on at least some of their trips from transportation hubs. Lots of canoeing and hiking trips available and into places few skilled photographers ever find themselves.
Also don't forget the wonderful multi-day programs run at some national parks, such as those by the Yellowstone Institute. The Yellowstone Institute's website suggests calling the registrar for mass transit information to arrive at the rally point for each program (may not be available to all program start locations?) They offer a lot of programs, including those that cater to photographers. I believe Glacier has a similar program as well as Rocky Mountain National Park, but check me on that. There may be similar programs at some other national parks. (I've taken a couple of five day Yellowstone Institute courses and they are fantastic.)
Well, it's getting late here and I'm recovering from a cold. Forgive the ramblings.... As Mark Twain once apologized at the end of a long letter to a friend that if he'd had more time to write, the letter would have been shorter.
I'll keep thinking about this more and pick my fellow ranger's brains some more. If I come up with other things, I'll let you know.
Hopefully, my ramblings and Cindy's great suggestions ideas by others as well.
Good luck with your trip planning....