Wights can cross, Benjen is a special case
Jon took a wight prisoner in the books and put it in the cells to learn from him/it, so the answer is yes: they can cross through the door, or at least the magic that awakens them can.
Monsters they may be, but they were men before they died. How much remains? The one I slew was intent on killing Lord Commander Mormont. Plainly he remembered who he was and where to find him ... My lord father used to tell me that a man must know his enemies. We understand little of the wights and less about the Others. We need to learn.
A Dance with Dragons, Chapter 39
On the other hand Jon put a body in the cell and it reawakened there. So maybe wights already awakened can't cross the doors.
Vahx mentioned in another question that one possibility is that we can consider the castle to be inside the wall. If that's so, the magic doesn't prevent wights from awakening; yet, as the question asks, we didn't see any awakening happening south.
My own speculative conclusion is that Benjen's case is probably different since he is not exactly a whight, and wights should be able to cross the door. I think the White Walkers are not crossing waiting to have a large enough army of wights to make the attempt. Then again, they could be searching/waiting for the Horn of Winter.
In S7E6 we finally see a wight captured behind the wall go beyond it, and not just someone awakened inside the wall.