@Daniel Suarez, as others have said, this a flashing problem, and @Jon H. is right that caulk alone is a Band-Aid. The pictures don't show this, but I think your house is like the one I bought 20+ years ago in California. It had extensive rot to all the trim in the back of the house. My guess is that you have horizontal lap siding, and that there is a trim board -- maybe a 2 x 4 or 2 x 6 -- above that door. Also, there is no flashing at the bottom of the last row of siding, adjacent to the trim board. So here's what happens: Water that hits the siding drains OK (with the possible exception of poorly sealed ends). The bottom of each siding plank overlaps the top of the one below. When the water reaches the bottom row of siding, however, it spreads out in the crack between that siding board and the trim board. Without flashing or adequate caulk, some water leaks behind the trim board, causing it to stay wet and rot, and some of the water leaks down onto the door frame.
The fix is this, if my guess is right about the construction of your house: First, remove the trim board above the door. You'll probably have to dig around in the caulk over some nail or screw heads to find the fasteners. If the trim board is rotten, cut a new piece. Then get a piece of Z-bar flashing, which vaguely resembles a Z when viewed from the end. One side of this Z flashing slips up underneath that bottom siding board. The main part covers the top of the trim board, and the other side of the Z extends down slightly in the front of the trim board. Now the water that flows down from the bottom siding board can't get behind the trim; it has to run out to the front of the siding. Don't caulk the top of the flashing to the siding; you want to allow drainage there. Also, if there is a chance of water draining from the bottom of the trim into the window frame, you need to caulk for that.