Most people arrange tours through their hotels or back in Bocas town. There are no true tour outfitters on the island and only a few casual ones. Up in the Hill (cell 6607-8962 and 6570-8277, www.upinthehill.com [1]) offers surfing lessons for US$40 for two hours, including a longboard. Surf tours are US$100/half day, US$150/full day.
Hostal Bastimentos (tel. 757-9053) rents surfboards for US$10 per day.
Pension Tío Tom (tel. 757-9831, tiotomscabin [at] gmail [dot] com) sometimes offers tours for around US$20–30 per person, and rents snorkeling gear (US$4 for snorkel, mask, and flippers), boogie boards (US$10), and booties (US$2).
The Dutch Pirate (Avenida Norte, cell 6567-1812, www.thedutchpirate.com [2]), over the water on the far eastern edge of town, is a scuba operator that also rents surfboards and other equipment. They also have an office in Bocas town.
Those staying in Old Bank [3] should consider contacting Pension Tío Tom (tel. 757-9831, tiotomscabin [at] gmail [dot] com) for tours of the island and the surrounding area. Tom Reichelt, half of the laid-back couple that owns the pension, is a landscape gardener and tree expert who is quietly passionate about this beautiful place. He leads tours of the interior of the island and has a special fondness for ones that involve searching for frogs or exploring caves that he says are the largest in Panama [4].
He had gotten a bit burned out on tours when I visited him in 2010, but he was planning on offering them again. Someone sufficiently enthusiastic about frogs can probably talk him into at least doing a frog-spotting trip. Expect to pay around US$10–25/person for tours, depending on the destination. Tom speaks English, German, and some Spanish, and he really knows his stuff.
Links:
[1] http://www.upinthehill.com
[2] http://www.thedutchpirate.com
[3] http://www.moon.com/destinations/panama/bocas-del-toro/isla-bastimentos/sights/old-bank
[4] http://www.moon.com/destinations/panama