The hyperbola $C$: $y^{2}-mx^{2}=3m (m > 0)$ can be rewritten as $\dfrac {y^{2}}{3m} - \dfrac {x^{2}}{3} = 1$,

Thus, we have $a^{2}=3m$, $b^{2}=3$, and $c^{2}=a^{2}+b^{2}=3m+3$.

Let $F(0, \sqrt {3m+3})$, the equation of one asymptote is $y= \sqrt {m}x$.

Hence, the distance from point $F$ to one asymptote of $C$ is $\dfrac {| \sqrt {3m+3}|}{ \sqrt {1+m}}= \sqrt {3}$.

Therefore, the answer is $\boxed{A}$.

The solution involves converting the hyperbola equation into standard form, determining the coordinates of the focus $F$ and the equation of one asymptote, and then using the formula for the distance from a point to a line to find the required distance. This problem tests understanding of hyperbola properties, the application of asymptote equations, and the use of the distance formula, and is considered a basic problem.