Masonry surfaces are, by nature, porous. That is why moisture problems are particularly predominant in basements and more often noticeable during rainfall. The solution is twofold.
First of all make sure the grade of your property slopes away from your house, so no rain has a chance to pond anywhere near the foundation. Then, use a good quality masonry waterproofer that controls the access of unwanted moisture by creating a barrier on the interior surface of your basement wall.
Masonry waterproofers, like thick paint, can be applied by brush or roller. Just be sure to look for a product that includes additives to resist mold and mildew, and includes a heavy-duty resin for maximum adhesion to the masonry surface.