Saltaire itself is a very family-oriented town, and there's not much around to stress you out. The town itself is limited to the beach, the library, a little market, and Fire Island Water Sports, which is a local, dog-friendly business!
There, you can get kayaks, paddle boards, and surfboards to ride the waves with your favorite-furry-friend (just look at their pictures, and you'll see that doggos can definitely learn to surf and paddle board!). Most of the sidewalks in this little town are boardwalks, making a walk on the beach even more pleasant. The best way to get around the town and to the beach is by bike (not many have a need for cars), so pack up some supplies on your bike and hop on the ferry over to the tiny town.
Make sure to also take a look at some of the historic houses on and around the beach - "elite residents have included the late fashion icon Liz Claiborne, director George Roy Hill, and former congresswoman Geraldine Ferraro." There are also two restaurants right off the beach, which you can take Fido to for a quick cool down and maybe even a snack or cocktail. Whatever you decide to do, you and your pup will definitely feel like you're hanging around in style, and will have the privacy that only small towns can provide!