This section of the house is slab on grade. There is a concrete step just in front of the door. The rotted wood riser fits between the house and the concrete step. I'm going to attempt replacing it. I think that the band joist and floor joists are ok. The door threshold itself is firm. When I replace the riser, should I caulk any or all of the highlighted seams? Is there anything else I should do besides attaching the new riser?
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.