I disagree even with the question being asked, as it is rather like asking "have you stopped beating your children?"
Jesus was born with a FULL human nature - the same human nature we have now. If that isn't true, that Christ didn't become as we are and we are still separated from God.
Yet if you, like Augustine, imply that our nature has "sin" within it - that we have a sin nature - then you necessarily make it impossible that Christ could take on such a fallen nature (as Christ is without sin).
If you accept the more Cappadocian view (Basil and the Gregories) and Athanasian view that our nature, after the fall, isn't fundamentally different but is, rather, subject to separation from God and thereby to death do to our own sin (and this is the same as the pre-fall nature, as implied by God's warning, except that because no sin had occured these consequences were not occuring), then Christ can have our literal nature - except that Christ doesn't chose to sin.
He became as we are, except without sin.
So yes, Christ had a "fallen" nature - or rather, He had the same nature as we do today (human nature after the fall). To say less is to undermine the Incarnation and destroy the Gospel. By His very goodness He redeemed that nature - but whatever is not assumed is not healed. If He didn't assume our nature, the Gospel is moot. But if He did assume our nature, then God (by His very changelessness) has transformed that state-of-separation into a means of communion. Even in our present nature, because of the Incarnation, we can be with God.
In Christ,
Macarius