The original sense dates from Old English - "full of care" - and this is the primary sense in Shakespeare. It means "anxious, worried" when Queen Isabel says of York: "full of careful business are his looks!" (Richard II, II.ii.75) or when Henry V soliloquises: "Let us our lives, our souls, Our debts, our careful wives ... lay on the King!" (Henry V, IV.i.224). It's a short step from there to "caring, provident," as when Lady Capulet says to Juliet, "thou hast a careful father, child" (Romeo and Juliet, III.v.107) or "protecting, watchful", as when Pericles says he "fledUnder the covering of a careful night" (Pericles, I.ii.81). And a further narrowing of sense takes us to "painstaking, serious-minded", when Feste reflects, "to be said an honest man and a good housekeeper goes as fairly as to say a careful man and a great scholar" (Twelfth Night, IV.ii.9).
David Crystal is author, with Ben Crystal, of Shakespeare's Words, published by Penguin