Under my front door, there is a wood board, but there is a 1/2 to 1 inch gap above the board and below the threshold. Is it a problem if water gets in there? I'm not sure if I should seal it. There's concrete visible in the gap, and it seems to be in line with the foundation.
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.