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Given:
Vertex, $A = (-1, 1)$
Focus, $S = (2, -3)$

Let the foot of the directrix on the axis be $Z(x_1, y_1)$.
Since the vertex $A$ is the midpoint of the segment joining focus $S$ and the foot of the directrix $Z$:
$\frac{x_1 + 2}{2} = -1 \implies x_1 + 2 = -2 \implies x_1 = -4$
$\frac{y_1 - 3}{2} = 1 \implies y_1 - 3 = 2 \implies y_1 = 5$
So, the coordinates of $Z$ are $(-4, 5)$.

The slope of the axis line $AS$ is:
$m_{AS} = \frac{-3 - 1}{2 - (-1)} = \frac{-4}{3} = -\frac{4}{3}$

Since the directrix is perpendicular to the axis of the parabola, the slope of the directrix is:
$m = -\frac{1}{m_{AS}} = \frac{3}{4}$

Therefore, the equation of the directrix passing through $Z(-4, 5)$ with slope $\frac{3}{4}$ is:
$y - 5 = \frac{3}{4}(x - (-4))$
$4(y - 5) = 3(x + 4)$
$4y - 20 = 3x + 12$
$3x - 4y + 32 = 0$