Bricks, Masonry, Asphalt and Concrete - outdoor stairway drainage

Hi, I have a concrete closed in stairway going down to my basement door. There is a small drainage hole at the bottom but my husband says it is not deep enough to keep the spot from flooding and then coming in through the door. How can I make this drainage hole deeper? Or should I just replace the rotten wood from the bottom of the doorframe, build it up a little bit and waterproof it. (It does have a overhang covering it)
The floor is starting to get covered in mildew so I want to fix the door before attempting the floor.

Any thoughts are appreciated.
Thanks,
Linda

Is the stairway covered or opened to the elements, the only way to fix it is rip out the floor, install new drain and pipe going to daylight, or to a pump.