"Cramped" openings are generally used by players who are good at piling up strong defenses for the opening and who hope to encourage overextension on White's part, in order to take advantage of it; or to fend it off till it has passed, and then muster an attack of their own. After 1. e4 e6 2. d4 d5, which is the common continuation of the French Defense, Black main drawback is, as you mentioned, his Queen Bishop, which cannot usually enter the play until the endgame. However, after ... c5, attacking the newborn White Pawn chain, Black's Pawn center may become threatening, and can grow into a major defensive asset, either forcing White to give up the center, or tying down White's pieces to defense of the center. The point is that Black has many attacking chances as well, and while his first moves appear conservative, even "cramping", he is merely being more patient than in the alternative openings. He uses more moves to get where he wants, but the extra moves can reap dividends when White comes to attack.
So a closed opening which is handled by a patient player holds much potential defensive advantage, and it is often played with a definite view toward strong reactions after White has burnt himself out trying to effect a successful attack.
Naturally, not all closed openings are effective. More than one kind of cramped opening exists: the kind that looks cramped, but is in reality compact and smooth (closed), and then there is also the kind that is really unworkable and impossible to manipulate (cramped). The latter turn into big mistakes, even worse than they look, especially when played by one who is more used to aggressive technique than the patient biding of time. It wouldn't be recommended to play these unless you are the patient type, or unless you can force yourself to play quietly and defensibly until it's time to strike.
Edited later to say that there are closed openings for White, as well.