Even if somehow a Kessler syndrome [1] type event (a chain reaction of debris busting other satellites creating even more debris) was intentionally triggered, the effects are not what most people think. Launches would remain perfectly safe simply because space is massive. What would happen is that certain orbital velocities would end up with an unacceptably high risk of collision over time, and so you wouldn't want to go into orbits that spend any significant amount of time at those velocities.
The neat thing about orbital mechanics is that your orbital altitude is determined 100% by your orbital velocity. Even in the case of an eccentric orbit, your velocity changes as you go from your furthest point to your closest point. A purely circularized orbit is an orbit where your velocity stays constant.
Extremely high energy debris would often end up escaping Earth's orbit and probably end up orbiting the Sun. And lower energy debris would often end up entering the atmosphere and burning up. So only fragments that remain in a sort of demented goldilocks zone would end up being dangerous. So in general I think the answer is - not much, especially in strikes of satellites near LEO. US, Russia, China, and India have all carried out live fire tests of anti-satellite weapons.