Scrub a wire brush over the crack in the basement wall, to remove loose chips and dirt. If the surrounding wall is exceptionally dirty, scrub it with a concrete cleanser and sponge. Allow the area to dry.
Insert concrete caulk into a caulk gun. Cut off the caulk tube's tip using a utility knife so the hole's diameter is approximately the same as the crack's width. Angle the cut slightly so it smooths the caulk bead as you apply it.
Place the tube's tip at the top of the crack. Pull the caulk gun trigger and slowly drag the tip down the crack. Hold the tip tight against the basement wall to force the caulk into the crack. Allow the caulk to dry, according to the manufacturer's directions.