How should I destroy a Mob Spawner?
To break one, use a pickaxe, preferably Iron level or higher. Note, once you destroy one, it will not drop as a block, so break these only if you'd like to get rid of them permanently.
While your opinion may differ, I find spawners to be excellent exp farms, and you can turn yours into a mob farm.
If you'd rather save your Mob Spawner, you need to block it off with light, similar to placing torches in a cave to ward off mob spawns. To de-activate a mob spawner, place torches around the spawner (at least an 8 block radius) to prevent mobs from spawning. Put them adjacent to the block itself.
Unless it is really in the middle of your building plans, you might want to keep it though, just in case you'd later on decide you like the whole mob trap idea.
Protip: If it's a skeleton spawner, just break a one block hole out of the wall, let them see you, and back away. The skeletons will start shooting and hit each other on accident. The ones caught in the crossfire will turn on their allies and kill each other off.
@Sonic42 Kinda late, but if you do that method, then you won't get any exp orbs out of it. Not worth my time.