To translate point \( B = (6, 4) \) down by 8 units, we need to subtract 8 from the y-coordinate of point B.
The original coordinates of point B are \( (6, 4) \).
After translating down by 8 units:
- The x-coordinate remains the same: \( 6 \)
- The y-coordinate is updated: \( 4 - 8 = -4 \)
So, the new coordinates of point \( B' \) after the translation are \( (6, -4) \).
Therefore, the coordinates of point \( B' \) are \( \boxed{(6, -4)} \).