Should I caulk a front door step riser when I replace it?
Definitely caulk under the threshold and at the ends. Whether you caulk the bottom partly depends on what's behind. If there's a drain plane back there, maybe leave the bottom open so water can weep out underneath. If not, seal it up. There's rarely any harm in caulking unless you trap water. 

I recommend urethane or pure silicone caulk. Acrylic latex just doesn't bond as well and shrinks horribly. Keep in mind that silicone isn't paintable, so choose white or clear, as appropriate (clear against the concrete and threshold). Urethane is amazing stuff, but it's difficult to work with, being so sticky. It doesn't let go of clothing or skin for a long time. 

I also recommend looking for a solid vinyl board for that location. You won't have to paint it and it'll resist decay much better than wood. As aaron says, if you do use wood, remember to at least prime (if not paint) the back, sides, and endgrain.