Sim,
I would be surprised if the sealer is going to be your answer.......that would suggest that the water is making it's way directly through the woodwork. Of course, it won't do any harm anyway, but I think your real answer lies in trying to find cracks and water paths.
These pre-fab sheds have a weakness at the corners where water can get past the vertical joint and find its way in. They also, obviously, have cracks all around the door.
If it hasn't got one, an old-fashioned pentice board above the door head (like a window cill, really, but above an opening), will keep quite a lot of weather off the door. A bit of 4x1 would be fine for this....and extend it past the sides of the opening by 2 or 3 inches. Give it a slight slope, and tuck a bit of damp course under the next board up and down over the pentice board, held with some staples.
You could also plant some cover strips over the face of the door at the jamb (it won't open if you try and do it on the hinge side........unless it is an inward opening door!).....a bit of say 2x3/4" , half on the door, half overlapping the wall.
Probably too late now, but I would always advise that the shed be lifted off the substrate by at least 6 inches.......water bounces up, and will easily saturate the bottom of a wooden building that is only an inch above the ground. Keep the ground clear and low all round the shed, and if you surround it by shingle you get a lot less splashing.
Post some photos........there may be something obvious I'm missing.
Mike