i suspect the fin is a modern innovation, probably started sometime around WWII...just a hunch...
none of these old boards appear to have fins
from wikipedia
Fins
The surfboard fin is a stabilizing strut fixed to the rear of the surfboard to prevent it from sliding sideways. In the early days, surfers would stabilize the board by hanging the toes of their back foot over the edge of the board and would steer by putting their foot in the water. The innovation of a skeg in 1936 —by either or both of Woody "Spider" Brown or Tom Blake— revolutionized surfing, allowing surfers to direct the board's momentum and providing more balance whilst turning.
The template of the modern surfboard fin was developed by George Greenough in the 1960s.<12> The single fin changed little until the late 70's, when a second was added and popularised by Australian Mark Richards.<13> The new twin fin set up allowed much more flowing carves to be performed. Mark Richards dominated the world competitive scene from 1979 to 1983. In 1981 another Australian was developing another set up which would again change the face of surfing. His name was Simon Anderson and by attaching a third fin, positioned centrally behind the twin fins, he created the thruster set up. Today, most surfboards still use the same arrangement with its popularity arising from the combined ability for carving turns and providing control and drive.
In the early 90's removable fin systems were developed and embraced. One of the most popular such fin systems is the FCS system, or Fin Control System. FCS is a more standardized system that allows fins to be easily removed or replaced, utilising set screws to hold the fins in place. These systems provided surfers with the ability to alter the riding characteristics of a surfboard, by changing the size and shape of fins used. This innovation opened the market to a range of fin designs, including single foiled fins, concave inside surfaces, and curved fins. Another variation of fin was later designed in the time frame known as the soul fin, a sleek bendable attachment.
Tunnel fins were invented in the 60's by Richard Deese,<14> and were found on longboards by multiple manufacturers of that era, including Dewey Weber. Bob Bolen aka 'the Greek' patented the Turbo Tunnel in the late 1990s. Since the mid 90's half tunnel fins have been used, mainly on very long hollow wooden surfboards such as those made in New Zealand by Roy Stewart.
Bullet Fins were invented in the 2005 by Ron Pettibone to increase surfboard hull planing and rail-to-rail transition speed. The patent-pending fins are based on 50 years of hydrodynamic research on the bulbous bow hull design. Just as with the bow of a ship, the traditional surfboard fin creates a wave as it displaces the water in its path. The resulting turbulence places drag on the surfboard.<15> The bulb of the Bullet Fin reduces this drag by creating a new (primary) fin wave in front of the original (secondary) wave. This new bulb wave is designed to be nearly 180 degrees out of phase with the original fin wave to subtract its turbulence thus reducing fin drag.
Winged fins are another type of surfboard fin, the genesis of which was America's Cup sailboat design.<16> The Starfin was designed in the 1980s by the America's Cup yacht designer, Ben Lexcen, who had designed the winged keel for the America's Cup boat, Australia II. The small thruster-sized fin, the RedTip 3D is manufactured by FCS.
Fins with winglets -- tiny wings -- are part of the Wavegrinder fin invented in 2005. See Wavegrinder patent. The purpose of winglets, as in airplane design, is to increase lift (horizontal turning force in the case of surfboard fins) while reducing drag, by reducing the fin-tip vortex.
http://en.wikipedia.org/wiki/Surfboard