The modern meanings are quite old: the "lazy" sense dates from the 14th century, and the "unoccupied" one from the 16th.
But by Shakespeare's time the word had developed a wide range of negative nuances, most of which are to be found in the plays. For example, when Parolles calls Bertram "a foolish idle boy", he means he is worthless (All's Well that Ends Well, IV.iii.210). When Hamlet tells Horatio that he "must be idle", he means he is going to appear mad (Hamlet, III.ii.100).
When Antonio talks about whores and knaves as being "idle", he means that they are frivolous or wanton (The Tempest, II.i.170). When Brabantio says, of his suit, "Mine's not an idle cause", he means that it is not trivial or unimportant (Othello, I.ii.95).
And when Mercutio describes dreams as being "children of an idle brain", he means that they are fanciful or foolish (Romeo and Juliet, I.iv.97).