Should I cover this gap under my front door?
Yep, caulk it and paint it with paintable caulk and exterior paint (after a good "surface prepping"). Use a dark shade to help hide foot scuffs. 

Don't caulk inside the metal threshold, it probably needs to be able weep water. Just caulk it where it meets the wood, and all around where the wood abuts the building, even if it's not cracked, yet.

The problem is not only with water infiltration, but air, as well with those lil' buggers.