We had to provide for our own transportation, since the Swim isn't booked through Disney. You'll get detailed transportation instructions when you book the Swim. They will send you a small package containing the needed info.
Basically, you'll take a taxi to Paradise Island ferry terminal($4/person). You'll catch a ferry there that will take you directly to the Swim.
Disney doesn't provide a list of "safe" taxis, that I'm aware of. Don't worry about getting taken advantage of price-wise for the taxi ride. We were told numerous times that it would be $4/person, and it was. I was all prepared to bargain with the driver, but never needed to.
You'll catch the taxi right after you get off the pier where the ship docks. Lots of hair-braiding offers there, so you may want to cover up your or your daughter's hair!
Also, the taxi will have to pay a $1 toll to cross a bridge during the ride. Our driver asked that someone in the taxi pay it. Have a dollar available. (I had all our money buried in a bag. My daughter, age six, wanted to play grown up and give the driver the dollar. Like any six-year-old, she took about five minutes to unzip her fanny pack, dig the bill out, rezip her pack, smooth out the bill ("it's all crinkly, daddy!") and hand it to the driver. Traffic is backing up, the other passengers are rolling their eyes, you get the picture.)
Also, we were told to arrive at the ferry terminal 1/2 hour to 45 minutes ahead of schedule. You don't need to. They apparently tell you that so you'll shop in the open-air market that's there. It was OK, but for me (a non-shopper) it was a waste of time.
Have Fun!