I think independent/dependent are the correct idiomatic choices. I might say it this way:
// Werd doch mal selbständig, Kind! - It's time you made yourself independent, boy/girl!
// Er ist noch sehr unselbständig. - He is still very dependent.

If you look at the LEO entries, the two sets of words have a range of meanings and are not always interchangeable. It seems that there is some overlap of meaning, though the sets of words approach the idea from different directions. (Abhängen - hang down, depend; selbständig - standing on one's own) This often happens with words, they overlap in meaning but are not totally synonymous.

As far as cryme's suggestions go, I don't find the first suggestion terribly idiomatic. Yes, "independent" seems a good translation for "selbständig", but the sentence just doesn't sound like something a native would say, at least to me. I think a completely different sentence would be used:
"It's time you stood on your own two feet, boy!"

The other sentence sounds idiomatic to me: He is still very dependent (on his parents).

Danke euch beiden.

@Robert:"It's time you stood on your own two feet, boy!"
Diesen Satz würde ich aber eher einem Teenager dann sagen, oder? Jemand der so langsam mal lernen sollte, für sich alleine zu sorgen und auch so langsam mal ausziehen könnte.

Was, wenn der Junge/ Mädchen noch kleiner ist und es eher um die kleinen Selbständigkeiten geht? Alleine seine Schultasche packen, alleine Schuhe binden, alleine sein Bett machen.... Würdest du es dann immer noch so formulieren?

You're right - that would be for a teenager or older. For someone younger, you would probably say something like
-You need to learn / It's time you learned (how) to do things for yourself!
-. . . (how) to do things on your own!
-. . . (how) to take care of yourself!

Vielen Dank, Robert! Hast mir sehr geholfen!