Are you sure that's where it's coming in, I suppose maybe strongly windblown rain could come thru the lower part of the door if it's really gappy but you do have a pretty good roof overhang there...
Weatherstrip around the sides and top of the door (actually you probably don't need it at the top, b/c of overhang). For the door threshold, either put in a proper metal threshold (from Home Depot) under the door, or make something similar yourself out of flashing (it won't work as well as a purchased one). And that should take care of it if that's really the problem.
Good luck, have fun,
Pat