How to hide the gap at the base of the stairs and the door, when installing laminate flooring

There are 3 potential moisture issues:

• standing water leaking in via the wall/footer
• moisture migrating through the slab
• high humidity

You need to first figure out which of those 3 is happening.

If the water is leaking in, that's an issue of gutters, grading, draining, etc. That needs to be fixed on the outside, first.

If moisture is migrating through the slab, then you may have a high water table. If that's the case, I wouldn't bother finishing the basement. If it's just a bit of moisture, you could get by with putting down plastic underneath the laminate flooring.

If it's high humidity, then this might be the gap issue. Engineered floors shouldn't expand/contract as much as real wood, but they still need plenty of space around the edge of the room to expand. Check for that. Otherwise, get yourself a quality dehumidifier and have it run 24/7.

I have just recently tackled the same problem with my own hallway. The solution I did was as follows. I cut a piece so that it would fit the door opening and a little more, so that it will go under the door frame. I have cut this piece so that on one side there is still the laminate "click" side, while the other is naturally cut off (this is piece 2 in the drawing). This way you can attach another laminate piece to this one. Next, I cut a narrow piece of laminate (piece 1) that can click on to piece 2. Finally, I have cut piece 3 so that it will butt against piece 2. Since there is no "click" side there, there is a small gap (if you can cut it accurately enough without leaving a gap, the better). This gap I have filled with wood glue mixed with sawdust from cutting the laminate boards. For my particular laminate it is a very close match to the color of the laminate, plus is it quite a small section (for me it was around an inch). From standing height it is barely noticeable, and can be improving by careful sanding. If you have something better as the filler, than by all means use it.

You can switch between pieces 1 and 3 (in terms of which one will have the "click" side, depending on the rest of you hallway laminate orientation.

For the assembly, piece 2 (or the one it will gone with) should have the "click" partially removed, so that it can be slid from the opposite room, under the door frame and into place, with the a line of glue holding the two pieces together. Then piece 1 can be clicked to place, and piece 3 clicked and glued.

Edit: here are some pictures. First, the glued part. In the picture it looks much worse than in real life (also ignore the lack of skirting board). The glue goes all around the piece since there were some gaps in all directions and as I said, it is barely noticeable from standing height:

Here the opposite side of the doorway, where the piece is clicked. Note that the central part (piece 2) goes well under the door frame:

And finally the entire door frame. Excuse the blurry shot, but I hope it is enough to give you an idea of my installation:

Edit 2: Following the OP's comment, I have an idea about how to put piece 2 in place if you can't slide it from the opposite room. This requires you to not have finished placing the boards in the hallway. You'll need to place some temporary boards up to the door in question, then measure and cut piece 2. Remove the temporary boards and slide piece 2 from the hallway. Then you can continue laying the boards as usual. To connect piece 2 with the real board before it, you still have to slide it out a little (out of the hallway, into the next room), lay the board and slide piece 2 back in. Sounds complicated, but I can't think of another approach.