If you're completely horizontal in the water and doing a proper "straight leg" fin kick, your legs are going to break the surface.
I find that I always have to modify my kick on the surface to keep my fins down.
Heavier fins may help you with this, but for me what I do is bend my knees (yes bend)
so that my thigh pushes down into the water.
My knee and foot are now below the water and all I do is kick slightly down (from below the knee) and finish by extending my entire leg behind me as I finish the kick.
You should get a two-punch in your kick, a small push when you kick, and another push as you extend and straighten your leg.
Make the kick nice, slow, and fluid. Also, you're going to have to do a little foot play as well. Keep your toes pointed down until you finish your downward kick, as you extend your leg, give the foot a slight push, as if your tippy-toeing.
(try not to exaggerate it either)
The bend is not really exaggerated. What I notice most people doing is that they bring their calf behind them to kick.
In this case what you're doing is bringing your thigh forward. (think climbing stairs, not riding a bicycle)
It requires a little more torque and it may stress your leg out too much if you're kicking like this constantly.
I only use this as a power kick, to glide across the surface, rather than kick across the surface.
Just my 2 cents, your experience may vary
But to go to the root of the question, a fin kick is far more effective if you keep the fin completely submerged under water.
However this doesn't mean it is energy efficient. If you're tiring yourself out too much trying to keep your fins under, is it really worth it?