Exactly, it wouldn't be hard to find a justification for him wearing a mask or a costume.
During WWII Captain America was sort of used to help the War effort, he was on posters, he went to schools to talk to children, he was probably in radio commercials, and news reels before movies. Cap was more than a soldier back then as well, he was a symbol to the soldiers, the enemy, and the people at home. Even if his identity wasn't a secret back then, although I think it should be back then, he would need a mask to help be a symbol.
Perhaps its like some others have suggested, during WWII the Super Soldier project was classified and so his identity would be too, so he wore a mask. So now that he is unfrozen people would still expect him to wear a mask because thats what Captain America looks like to them.
Or maybe Captain America is meant to be more of a symbol than a man, in the eyes of Steve Rogers of SHIELD or something, and so he wears a mask to give Captain America his own face and his own identity.
One story I read Steve said he died that day and was reborn as Captain America, so maybe its like Nolan's Batman. the Captain America mask is Steve's real face.
There would not be a problem figuring out why he would still wear a mask even though the public knows who Captain America is (he is Steve Rogers). It would just have to depend on the story being told.
If you make his identity known then you sort of loose the whole Kent, Parker, thing were he has a secret life and a girlfriend and a job as a comic artist, and friends who had no idea who Steve really is.
However it is more realistic to think that his identity would be sort of known to the public. Even if its just a matter of public record once the Super Soldier Project is declassified.
Again, it also makes sense that his identity would be secret. To keep those around him safe and to allow him to have a life outside of Captain America.
But again, you would make his story just another superhero story where the hero has a secret life. making his identity known, and making him more of a public servant or government agent, makes his story different than Spider-Man's, Superman's, Batman's, even Iron Man's.
Perhaps his identity being a secret can be a major issue for the character, he has no reason to keep it secret and feels like the public should know who he is, but then he wants Cap to be a symbol of America and so on. At the end of the movie on live TV Cap can remove his mask, look straight into the camera, and say "I am Steve Rogers, and I am Captain America!"
Like he did in that Cassaday illustrated comic.