Header question for newly created exterior basement door
When you make a window or door opening, you will remove one or more studs in the door opening in the wall.  The header has to carry the load that was previously carried by the studs you remove, transferring it to the king studs on either side of the door.  

Since the door is stacked, aligned under a window above, the header for the window opening has already transferred  the load above the window to the studs on the sides;  the stud removed for the door probably wasn't carrying much load.  It was carrying the weight of the window and the cripple wall below the window, which isn't much.  Depending on the direction the floor joists, run, that stud was also the supporting the floor joists resting on the top plate above the stud.  

I'd say if you have 


A doubled top plate above  the door is in
Joists not supported by the wall with the door 
Just one stud removed for door opening  


Then it's not likely you'll see any problem there. But you'd normally put something there anyway, if only for a nailing surface for the trim etc., and to avoid debates about it.  

In any event, I don't see how failing to install the header would lead to cracks under the door.  If the header is missing, the wall above the door opening may sag, the floor above the door opening may sag, but no additional load winds up on the floor around the door opening.  The door frame pretty much floats between the king studs.  If it was bearing any weight, it would probably deform and the door wouldn't open or shut properly.