This Old House recommends using a power auger, or snake, to remove clogs in the basement drain. A hand snake can be used to remove blockages if a power auger is not available.
Determine whether the drain has a clean-out plug by checking around the perimeter of the drain basin. If a plug is located, remove the cover and turn the power auger on. Feed the cable into the clean-out plug opening slowly, and stop when cable meets resistance. The resistance is typically caused by the blockage in the drain. Rotate the cable to catch the blockage, then slowly remove the cable. It may be necessary to move the cable up and down in the pipe if the blockage is particularly large or if it is lodged in the drain pipe.
If a clean-out plug is not present, remove the drain cover and feed the cable into the opening left by removing the drain cover until the cable meets resistance. This Old House recommends twisting the cable clockwise until locating the blockage, then switching to counterclockwise rotation on the power auger to catch and remove the blockage.
It is also possible to remove a blockage in the basement drain using a snake by hand. Insert the snake into the drain while rotating the cable. After locating the blockage, twist the cable to catch the blockage and pull it out of the drain. Clearing a basement drain by hand may require snaking the drain several times before the blockage is completely removed.