Your only choices are to walk (the island from end to end is about 3.8 miles/6.2 km), or rent a cycle, scooter, or off-road golf cart (recommended).
Walking is not recommended. It is extremely hot and dry on Salt Cay, and there are very few trees on the island (no shade), with the exception of a few non-native casurinas near Balfour town. If you decide to walk, make sure you bring plenty of water and a hat.
Several of the villas and guest houses include a bicycle, and local businesses rent them for day trippers. Again, this is a very hot and dry island and cycling will not be for everyone, although the island itself is rather small so the distances travelled are not too far.
Two local companies rent off-road golf carts for rent for around $80/day including gas (2013 price). This is a fun and highly recommended way for getting around the island. Despite having a sun roof, you will still get sun-burn so make sure you bring sun cream and a hat. Don't worry about gas as the tank will last you the full day.
Scooters are also available for rent, but we think you'd probably enjoy the off-road golf carts the most.