ASA still requires an attempt by the batter to avoid being hit. This is regardless of if the ball is in the batter's box or not. Some sanctioning bodies have removed this obligation of the batter on a pitch inside the batter's box, but ASA is not one of them.
The quality of the "attempt to avoid" is, of course, left to the umpire's judgment. On a ball in the box, you do tend to give the batter the benefit of the doubt. Even a slight flinch can be enough of an effort. Some pitches are, by their very nature, unavoidable. A pitch in the box can freeze the batter, especially if the batter is mid-stride.
If the umpire judges that the batter purposely allowed the ball to hit her, then the pitch is called a ball or strike depending on its relation to the strike zone and no base award is given.