Step 1: Concept
Precipitation requires ionic product to exceed $K_{sp}$.
Step 2: Analysis
With $K_{sp} = 10^{-8}$ and $[A^{+}] = 10^{-3}$, $[B^{-}]$ threshold is $10^{-5}$.
Step 3: Conclusion
It precipitates when $[B^{-}]$ is more than $10^{-5} M$.
Final Answer: (C)