Effective use of Silence depends on a coordinated strategy involving battle field positioning/control. There are a number of ways to limit/control caster movement, but here are some of the more effective ones:
- Cast Silence after your melee characters are in range of the casters (thereby limiting their escape/provoking OA). This tactic is especially deadly if your melee characters have taken the Sentinel + Mage Slayer feats. If not, when facing a very dangerous caster, one melee character should consider grappling. Grappling a caster inside Silence will severely limit their spellcasting. Further, most (if not all) 'escape spells' (Dimension Door, Misty Step, etc.) have Verbal components, so, if they can't walk out, they can't cast out either.
- Coordinate with another caster to first cast a spell that limits movement, either by immobilising enemies or creating difficult terrain.
- Coordinate with PCs that can move enemies around (Warlock with Repelling Blast, Druid with Thorn Whip, etc.) to move casters back into the zone (especially useful if combined with 1: move them next to your melee allies). Characters with the Charger feat can also shove casters back.
If you provide more specifics about the scenario, I can develop the answer a bit more.