Grammatically all of them are correct. There are slight connotation differences among them, though. A. 바쁜가요? B. 바쁜거야? A and B have the same meaning but A is in a polite form and B is in a familiar form. The basic verb form is 바쁜것이다. 바쁜가요 = 바쁜것인가요 바쁜거야 = 바쁜것이야 -ㄴ is emphatic. If someone asks you "당신 오늘 바쁜가요? (Are you 'busy' today?)", he or she wants to know if being busy is your state. If someone asks you "당신 오늘은 바빠요? (Are you busy 'today'?)", he or she wants to know if today is a busy day for you. If someone asks you "당신은 오늘 바빠요? (Are 'you' busy today?)", he or she wishes to know if it is you who is busy. 바빠요? This is a formal form without any emphatic case ending.

"바쁜가요?" = slightly old-fashioned, I heard middle-aged people talk like this. "바빠요?" = standard, common "바쁜 거야?" = more indirect. Could be stronger as in "그게 바쁜 거야?".