Seek out any cracks along the basement wall. Cracks will most likely form either at the top or bottom of the wall. Sometimes these cracks are large and quite obvious. Other times they're tiny and difficult to find. For best results, use a flashlight to provide light, even if your basement is lit.
Watch out for a white-hued deposit in any area of your basement (especially the walls). This deposit, called efflorescence is a harmless deposit that develops on the surface of concrete. It is a telltale sign of leaks. If you notice this in your basement, it's safe to assume the crack is somewhere nearby.
Look for any paint that has peeled, especially if your basement is finished. Bubbling, peeling or otherwise disturbed paint is a sign of water damage.
Note any spots on your basement wall, ceiling or floor that have accumulated mold or mildew. Where there is mold or mildew there is (or was) water.
Check the floor for any lifted floor tiles or even rust around metal appliances. These, too, are signs that water is present, and any leak you have will be close.
Inspect both the water system and plumbing in your home to see if you have any hidden leaks. Look at water supply lines, sinks and pipes for any leaks. Running toilets can also be the source of your basement leak.
Remove any false ceiling tiles (if possible) for leaks that may occur in the pipes that are stashed away. Remember that the source of leaks that travel along pipes may not be directly above the water stain on the ceiling tile.
Contact a professional if you still cannot find the source of a leak (or to fix the leak you have found, if necessary). If you cannot find the leak, it's possible you have a pipe leak lurking behind your wall.