Because ladies and men like smooth faces.
Because it's a distinguishable feature that's easily identifiable. Ollie Queen's goatee should've been the first clue that he was Green Arrow; current, clean shaven Ollie has anonymity due to his lack of a beard.
Also because of these reasons.
Part of the reason is because Western society has been conditioned to shave off all facial hair. A practical reason is because a person's beard may be shaved a certain way that makes them identifiable and that prevents concealment of a secret identity. Hair and beards can also make you more vulnerable in battle.
Although, Superman and Green Lantern have both proven how little one has to do to have a secret identity.
Maybe Bucky isn't a "Super-hero", but he's certainly a hero. And hey, he often has a beard, and as Captain America, maybe he was shaving more often, but the beard was also visible from time to time (as Winter Soldier he seems not to care that much if he;s shaven or not). So, hurray for the beards!
Typically, when a hero appears with a beard, it is an easy way for artists to show that they are so stressed that they no longer care how they look. Thematically speaking, I think a beard conjures an image of ruggedness and survival-oriented thinking that is at odds with the preservation of order most heroes typically are associated with. The image of a bearded lumberjack is very different from a shaved soldier, partially because the discipline to shave and maintain a standard has more to do with preserving society according to the last few generations of Western imagery. It is why bad guys in stories have beards, while good guys are often clean shaven.
On a more pragmatic level: I tend to think that artists would hate drawing bears on a regular basis; heck, most artists don't even like drawing the textured hair of african hair types or afros. It is also easier to make clean shaven characters look generically handsome than bearded characters.
Also: Beards are gross. There, I said it. They are called soupcatchers for a reason. Plus the main benefit of frequent shaving is the exfoliation that occurs; men who shave often have better skin because the shaving regiment forces men to take care of their face every day or two, without needing to go to a spa, or even think about it. (Just note that this is a humorous aside).
However, I'll admit that there is a big difference between this:
It is rather perplexing that of nearly all the big name heroes out there, Thor the Norse God, is clean shaven. And I say that because culturally amongst the Norse a beard was a sign of manhood, as in if you didn't have one you were either a child, thrall or a woman.