When a homeowner has a wet-basement problem, an attempt frequently is made to waterproof the walls from the inside. The solution, however, is often outside the walls.
As an example, take concrete-block basement walls that have a dampness problem in one corner. Outside the damp corner there is a downspout that drains into a pipe running along the outer wall and eventually draining into the street.
Although various coatings are applied to the wall where the dampness occurs, nothing may seem to help. Better results might be obtained by concentrating on the slope of the ground outside the wall and the condition of the rain guters and downspout. The problem actually may be than an unusual amount of rain water is accumulating in the ground outside the corner of your house, and the water then is working its way through the wall.
Many basements take on water simply because the ground around the foundation doesn't slope away from the house so that water runs off instead of accumulating. In other cases, a downspout dumps water directly on porous ground near the foundation instead of carrying it away from the house or clogged rain gutters or downspouts are causing water to pour down around the foundation.
If the ground around a foundation is level or slopes toward the house, it's wise to add some soil, reslope it and plant ground cover or grass to form a runoff belt around the house.
Concrete splash pans, sold at most home centers and lumber yards, can carry off water dumped too close to a wall by a downspout. A downspout also can be extended horizontally so that water is taken away from a wall -- add an elbow and some pipe to the bottom of the spout.
In your case, there might be an overflow of water at the joint where the downspout meets the drain pipe. Your description seems to indicate that the joint is above ground, so I suggest take a look during a heavy rain and see if this is the source of the water that is entering the wall. If the joint is overflowing, there is probably an obstruction in the drain pipe, possibly an accumulation of leaves.
To get a permanent solution, you might have to disassemble the joint and run a plumber's "snake" through the drain pipe to open it. Also install a leaf strainer -- an inexpensive wire device available at hardware stores -- at the top of the downspout, or install special wire or plastic screening over the gutters to keep out leaves and debris. If disassembling the joint seems impractical at this point, you can probably get some relief by installing a splash pan under the overflowing joint.