How does the widespread study of environmental science affect government policy?
Awareness has a big impact.
Most people that oppose environmentally favorable policies don't know about them, just as most people who deny evolution don't really understand what it is. If you're taking environmental science, you're likely learning that many of the measures to reduce our impact on the environment are also politically beneficial and a boon to our economies.
Additionally, you learn about a great many impacts we have on the environment as well as ourselves through unsustainable policies and actions that the unlearned person probably isn't aware of.
As more people study environmental science, it will likely influence how people vote, who runs for government positions, and thus, what policies are proposed and passed by the government.
I read a study recently that said that "including scientists and scientific information will improve the quality of complex policy decisions." (Source at the bottom) Though this specifically concerns scientists, it is reasonable to assert that more people in possession of scientific information (in this case environmental science) will change governmental policy in regards to the environment.