Translation:Are you busy?
If you add 很, 很 here is more like adv, which means are you very busy/extremely busy? In some context, 你很忙吗？can be regarded as irony. For example, my boss ask me to complete this sheet today, but I reply, no, I don't have time. Then my boss can have a sarcastic comment like 你很忙吗？means it just take you 5 min, are you really that busy?