There are a number of things that should be done here.
First, looking at preventing other countries from acquiring the weapons, the main thing that can be done is to create treaties and other agreements that will make it less likely that countries will acquire such weapons. Our experience with nuclear proliferation tells us that this is not fool-proof, but it is the most realistic way of approaching the issue. If countries that have signed such agreements start developing weapons, it is at least possible that international sentiment would allow strong action to be taken against them.
Second, if countries do acquire such weapons, we should prevent them from using them through treaties and through deterrents. Treaties will bring legal and moral pressure on the countries not to use the weapons. If we also state clearly that we will take harsh measures (presumably including war) against any country using biological weapons, we can hopefully deter them by making it plain that using such weapons would not be in their long term national interests.