There are no substantial waves when Batman & Joker enter or exit the ocean, but they are surfing on waves in between.
Mere moments after exiting the ocean Batman and Joker are both completely dry.
Batman tells Robin to go back to Wayne Manor and come back as Dick Grayson because he heeds the latter's surfboard. Dick arrives and Batman tells him to leave his surfboard, which is red with a white stripe, outside the "Hang Five". After the challenge is made and accepted Batman and Joker come running out to the beach and Batman has a yellow surfboard with a Batman logo on it.
Caesar Romero's mustache can clearly be seen under his Joker makeup.
In the Batcopter scene, Robin's stunt double can be seen clearly. He has light, receding hair and a muscular build.
When Batman and Robin are being turned into "human surfboards" it is obvious that after having been covered in whatever paste was being used as a prop, nothing is coming out of the henchman's hose.