Some sort of large rocket that would fly to the right orbit height, and orient itself in a manner to ensure it's shaped charge explosives would send maximal debris (ball bearings of the right size) in directions azimuthal to the planet. How many satellites could you knock out if you really tried?
Quite an optimization problem for the secret skunkworks folks.
With a single launch scattering a lot of debris?
You could reliably take out ONE satellite. The clouds of debris creating a cascade of destruction as depicted in the movie Gravity are entirely fictional.
Edit to add:
The easier and cheaper villainous way to deny spaceflight to someone is just to shoot missiles at their spacecraft.
Missiles can be pretty expensive, and if you go far downscale on cost and capabilities, you'll need to recruit fanatics who are willing to get caught every time a missile is fired at a spacecraft.
On the other hand, if the fleet of BFRs grows in numbers, then 100 flights of a BFR might be pretty inexpensive for a military operation that affects all strategic theaters at once.
The chief advantage of missiles, would be the ability to not shoot down your own rockets.