You can get a bus, to DHS, as early as 6:30am. The boats won't be running, so you would have to walk over to the Dolphin.
The two resorts are very close...I would take a cab. The cost should be around $10, tip included. At least that's what it was when we went from the YC to POP...and it takes less than 10 mins.
If you don't want to spend the money on a cab, take a bus to DHS and then walk over to the Dophin...about 20 mins on foot. If the boats are up and running (about an hour prior to park opening), you can head to the boat dock, at DHS, and take the boat. If there is a boat there, or you see one heading in, get on. It will take you about 15 mins, or less, to get to the S/D dock.