Let the side length of the square iron sheet be \( s \).
When the sheet is rolled into a cylinder, the circumference of the base of the cylinder equals the side of the square:
\[ \text{Circumference} = s \]
The circumference of a cylinder is given by:
\[ 2\pi r = s \]
Solving for the diameter \( D \):
\[ D = 2r = \frac{s}{\pi} \]
The ratio between the diameter and the side of the square is:
\[ \frac{D}{s} = \frac{\frac{s}{\pi}}{s} = \frac{1}{\pi} \]
Thus, the correct answer is \( \frac{1}{\pi} \) (Option B).