Step 1: Understand what is being asked.
We are told $x=\sqrt{3}$ satisfies $ax^2+\sqrt{3}x-12=0$, and we need to find $a$. Instead of substituting and simplifying the whole equation in one go, let us first isolate $a$ algebraically and then plug in the value of $x$.
Step 2: Isolate $a$ in the equation.
Starting from $ax^2 + \sqrt{3}x - 12 = 0$, move every other term to the right side:
\[ ax^2 = 12 - \sqrt{3}x \]
Divide both sides by $x^2$ (valid since $x=\sqrt{3}\neq 0$):
\[ a = \frac{12 - \sqrt{3}x}{x^2} \]
Step 3: Substitute $x=\sqrt{3}$ into this formula for $a$.
\[ a = \frac{12 - \sqrt{3}\cdot\sqrt{3}}{(\sqrt{3})^2} \]
Since $\sqrt{3}\cdot\sqrt{3}=3$ and $(\sqrt{3})^2=3$:
\[ a = \frac{12-3}{3} = \frac{9}{3} \]
Step 4: Simplify.
\[ a = 3 \]
Final Answer:
The value of $a$ is 3, matching option (A).
\[ \boxed{a=3} \]