Let's analyze each statement about the function $g(x) = 3|x+1|$ carefully.

Step 1: Understanding the structure of the function

The function $g(x) = 3|x + 1|$ is an absolute value function, which creates a "V"-shaped graph. The graph will have two distinct parts: one part will be decreasing and the other increasing. The vertex occurs where the expression inside the absolute value is zero, i.e., at $x = -1$.

Step 2: Analyze each statement

A. $g(x)$ has a y-intercept at $(0, 3)$.

To find the y-intercept, we substitute $x = 0$ into the function:

$g(0) = 3|0 + 1| = 3 \times 1 = 3$

Thus, the y-intercept is $(0, 3)$, so this statement is correct.

B. $g(x)$ has one x-intercept at $(-13, 0)$.

To find the x-intercept, set $g(x) = 0$:

$3|x + 1| = 0 \quad \Rightarrow \quad |x + 1| = 0 \quad \Rightarrow \quad x + 1 = 0 \quad \Rightarrow \quad x = -1$

Thus, the x-intercept is at $(-1, 0)$, not $(-13, 0)$. Therefore, this statement is incorrect.

C. $g(x)$ is decreasing over the interval $\{x \mid x < -13\}$.

The function $g(x) = 3|x+1|$ is decreasing for $x < -1$ because for values less than $-1$, the slope of the absolute value function is