The original terms goofy-foot and goofy-footers (later shortened to goofy) appear to have become popular in surfing during the early sixties. I agree with FumbleFingers: it's likely both the surfing term and Disney's Goofy character comes from the earlier goofy meaning of stupid, silly, daft.
In fact, after watching the 1937 Disney animation Hawaiian Holiday, Goofy surfs with whichever foot forward makes him face us. He attempts to surf three times. The first two are unsuccessful and he can't stand up on the board. The third time is (more) successful: first he surfs left-foot forward (regular stance) towards the right, so his body is facing us. Then he turns and surfs right-foot forward (goofy stance) towards the left, again so his body is facing us.
The earliest instance I found in print is Desmond Muirhead's 1962 Surfing in Hawaii: a personal memoir:
People who put their right foot forward are called 'goofy foots'.
2007's The Concise New Partridge Dictionary of Slang and Unconventional English agrees with 1962:
goofy foot noun a surfer who surfs with the right foot forward. Most surfers surf with their left foot forward. AUSTRALIA 1962
A 1963 Paradise of the Pacific magazine defines some surfing terms:
As or the language, it is probably altogether as intelligible to the unpractised ear as Kurdish might be in Kansas City. In the lexicon in use by the cult, a surfer is a cork top, beginners are gremlins or kooks, a loudmouth a hodad; a goofy foot, a strange type; a hot dogger, an expert.
A May 1963 Billboard picks out Shean and Jenkins with their Goofy Footer Ho-Dad single (listen on YouTube) as a winner in their novelty spotlight of the week, selected for "potential to become top sellers".
Two very funny sides that could attract play and sales. ... Flip is somewhat on the surf kick with a sort of beatnik poeatry narrative. Funny material, well carried off.
A June 1963 Billboard magazine lists a record by The Lively Ones called Goofy Foot as a four-star single (listen on YouTube).
The four-star rating is awarded to new singles with sufficient commerical potential in their respecitive categories to merit being stacked by dealers, one-stops and rack jobbers handling that category.
These show the term was becoming more popular and widespread.
An early etymology is suggested by the 1970 Studies in English by the University of Cape Town's Department of English says:
Surfers who have a right foot forward stance are known as goofy-footers, or simply goofy surfers. This is doubtless derived from the older American expression "goofy", which means "ridiculous, silly, . . . nutty".