You don't have to be on a board, or even in the water, for a surfing workout routine. You should consider four key elements when thinking about a routine.
Surfing Workout Routine - Stamina
Being out on a board for hours is going to take endurance. Even an hour, or half hour, can exhaust those just starting out. Simple exercises done for extended periods of time can help improve you stamina. Things like jogging or jumping rope for twenty minutes can help quite a bit. Remember to keep a steady pace through the whole twenty minutes. It's all about training your body to function just as well at the end of that time as when you started the exercise.
Chest and Back
The muscles in your chest and back are going to play key roles in surfing. Because of this you should incorporate them into your surfing workout routine. Consider rowing, push-ups, and pull-ups. These will target the muscles you are most likely to use.
Surfing Workout Routine - Legs
You will need some strength in your legs to ride a surf board. However, the stamina in your legs is also going to be important to keep you on the board. If you can't hold your stance, chances are you will end up falling off a lot. Consider doing extra lunges with both legs. Add ten to the number you normally do, or start with ten if you don't have them in your routine. Lunges not only build strength, but the stretch the muscles as well as using most of the muscles in your legs.
Try incorporating a stability ball into your surfing workout routine to help with balance. Something as simple as standing on one leg, and then the other, could help too. You will need balance to stay upright on the board. Everyone gets sick of wiping out, and improving your balance increases your chances of riding the wave.