Is being busy a positive attribute? These days it probably is. By 'these days' I mean at least the last century or so. I can remember being told off at school for day-dreaming or looking out of the window, whereas if I had picked up a pencil and pretended to be writing something, and thus had looked busy, I would probably have been thought a model pupil.
Words change in meaning, though, and social attitudes change over time, and being busy wasn't always considered a virtue. From around the 14th century, in addition to the meanings "active, assiduous, diligent, industrious", and "constantly or habitually occupied; always active or employed; having a great deal to do", busy also had the negative sense of "active in what does not concern one; prying, inquisitive, gossiping; meddlesome, officious, interfering" (OED). In Shakespeare's Othello, Emilia, the wife of Iago, uses busy in this sense:
I will be hang'd, if some eternal villain,
Some busy and insinuating rogue,
Some cogging, cozening slave, to get some office,
Have not devised this slander; I'll be hang'd else.
In modern English, this 'meddling' sense of busy is retained in busybody, and also perhaps in the British slang usage of a busy being a police officer or detective (often seen as the busies and also sometimes written bizzy/bizzies).
Busy is an old word; it has been around since the early Old English period, when it was spelled bisig or bysig. The Old English y sounded similar to the modern French u, as in tu. The u in busy crept in in Middle English, around the 16th century. A similar thing happened with the vowel of the word bury, which was spelled byrgan or byrigan in Old English (where the g would have sounded like the modern English y, phonetic symbol [j]).