Very short walk, I'd say about the equivalent of entering the Magic Kingdom and walking to Cinderella's Castle, if not shorter.
The Dolphin and the Swan use the exact same transportation as any of Disney's Epcot Area resorts. Friendship boats to/from Epcot and MGM, Disney busses to/from Magic Kingdom, Animal Kingdom, Downtown Disney and Disney's waterparks.
The boats stop at Swan/Dolphin (shared dock), Yacht/Beach Club (shared dock)and the Boardwalk each and everytime they travel to/from Epcot and MGM.
Hint...when coming from Epcot on the boat, if you're in a hurry, get off the boat at the Boardwalk and walk to the Dolphin. You'll beat the boat, because it goes back across the lake to the Yacht/Beach Club before heading to the Swan/Dolphin. When leaving MGM, you'll be the 1st stop.
The busses are sometimes shared with the same hotels, sometimes they are not. I've been at the Swan when our bus to the Magic Kingdom stopped at the Dolphin, Yacht Club, Beach Club, and the Boardwalk (yep, that's 5 stops) before heading to the park, and I've also experienced the bus only stopping at the Swan and then to the Dolphin before heading out.
Have a great time and enjoy your $$$ savings.