Inspect the hinges on the inside of the doors. If the doors get much use or rough use, then the hinges can become loose, causing the doors to match up improperly. If you see any loose screws at the hinges, tighten them and test the door.
Clean the sill where the leaks are coming from with a wet sponge and some mild detergent. Dry it completely and apply a thick layer of weather stripping to the sill. Most weather stripping has a simple peel-away sticky back and can be placed on the sill where it meets the door. The weather stripping will compress and fill the gap, sealing out any air or water.
Remove the doors and then make certain that the hinges and positioning are completely level and flush before reinstalling if all else fails.