How do I build down?

Place water and lava on adjacent sides of the block.

Top view:

  D L
  W

When they flow it creates a column of cobble you can dig through.

Side view:

  W D L
↓ ↓ C ↓ ↓
↓ ↓ C ↓ ↓
↓ ↓ C ↓ ↓
↓ ↓ C ↓ ↓

D = Dirt
C = Cobblestone
W = Water
L = Lava
↓ = Flowing water/lava

Use water scaffolding.

Check out this video at about 10:00 to see water scaffolding in use.

The lava/water method seems easiest, but if you don't have lava, you could also use a piston to push a bunch of blocks down, one at a time. Then mine safely down at that same spot. Make sure not to mine too far