There is no constant swearing in the game.
The guy says poop passage in the tutorial video, nothing during gameplay whatsoever aside from some laughing (comments from npcs are pretty tame), says an end taunt similar to Duke after beating that section of the score attack stage ("Now thats what my old man called fuckin fools up"), says "oh shit" when looking at a huge boss during the end demo trailer, and of course dick-tits at the end of the demo trailer.
Thats it. During gameplay? Not a word.
EDIT: I think the word dick tits is stupid too, but after actually playing the demo, I had fun with the game and its over the top, nineties style atmosphere. Its just that people keep saying there is cursing all over the place, when it really is limited to a few words before and after the demo gameplay.
Another heads up: Even though the gameplay is still fairly score based, the story mode is supposed to be challenging and more focused on progression than racking up points. Echo (score attack mode) is more or less the Mad World kill with skill focused section.