Is the sill sloped properly? Is that exterior tile pushing the sill up and making it almost flat as it appears in the picture? If so then the water will not run off like it is designed to do?

So it's all screen and rain can just come in? Is the bottom solid border higher than the concrete? And there's nowhere for the water to drain out? That just seems like a horrible design. It's basically a bucket. Or is there a gap at the bottom of the border and the concrete and the water is supposed to run out there? Is the problem that the concrete pad isn't sloped properly?

It could be many issues. With no specifics, there will probably not be many helpful answers. Is the water coming in from the bottom, the sides, or the top? Is it leaking in around a glass? Is it making the floor wet, or the wall board wet, or what? What kind of door is it? A storm door, a solid door? Is it under an overhang, or exposed to the side of the house? Is there a concrete patio or a wooden porch, or deck, or what outside the door? Is it an old door? Is the door warped? Could it be that you might have a clogged gutter that is allowing water to cascade down the side of the house and come in through the door? Maybe? Without some details, it is anybody's guess. Your bug problem and your leaking problem may, or may not, be related.

Water flows downhill. It could be your upper windows are not adequately flashed. It's always useful to see what's happening when it is raining. Go out in the next windy downpour and look. We had mystery water leaking and crawling around under the porch in a downpour --- utterly miserable --- made everything blazingly clear. We fixed the problem and have had no more water in the basement. The fact that the door has an awning suggests flashing is an issue.