The altars do not spread corruption, unless you break them. When you break them they cause a small amount of corruption(or hallow) to appear randomly on the map somewhere.

What happens when you destroy a demon altar?

Destroying a Demon Altar is directly associated with Wraiths spawning. Demon Altars do not spawn or respawn; there are a set number when the world is generated. Demon Altars can not be picked up after their destruction, as they simply vanish.

Should I break all the altars Terraria?

Altars are needed for crafting certain items, hence destroying every altar in a world is not recommended. It is not strictly necessary to smash an altar; all ores obtained by doing so can also be obtained by fishing up crates, allowing access to all ores and their counterparts in one world.