How big do artificial biomes need to be to spawn monsters?

The Terraria wiki does have an answer on how big many of these biomes have to be:

http://terraria.gamepedia.com/Guide:Artificial_biomes

For example, snow biome requires 300 snow blocks, ice blocks, or snow bricks.

Generally, you'll want to keep adding blocks until the background changes.

You can also try a Water Candle to increase monster spawn rate.

Monsters only spawn in a certain distance in tiles from the player i.e. between 62 - 84 blocks to the left or right of the player and between 35 - 47 up and down. Its not enough that the biome has to be off screen but it has to be in between these distances. So either make the biomes bigger i.e. 168 * 94 blocks with the player in the centre or dig out the middle and have the edges the biome you want.

I can't even get the background or music to change. But that didn't worry me and it shouldn't worry you because to my surprise in my artificial snow biome failure a zombie eskimo spawned.
Basically it doesn't matter what size the biome is, mobs will still spawn just slower. However the bigger it is the more it will spawn monsters.
Hope this helped.

Review the Terraria Wiki for the exact specifications for creating an artificial biome. If you do encounter any anomalies, ie. enemies from different biomes spawning, it's because most biomes need only 1 block to spawn enemies and plants. For example, you need a minimum of 1000 blocks to create a fully artificial desert biome, complete with background and music. However, 1 block of sand can still spawn waterleafs, antlions, vultures, and even hardmode NPCs.