If the dryer vent pipe gets too cold, you might end up with a leak where the vent connects to the dryer. Warm, moist air normally goes from the dryer to the vent, where it's carried outside of the house so that your dryer doesn't overheat. However, if the vent pipe gets cold, the moisture inside the pipe condenses, turning into liquid. If the vent pipe isn't sealed properly, the water then leaks out of the pipe.
If cold air is leaking into the house from the dryer vent, this contributes to the vent pipe getting cold, which can then lead to water leaks. Check the hole around your exhaust pipe. Put your hand over the exhaust pipe and see whether you feel cold air coming in. If you find an air leak, put caulking around the hole where your exhaust pipe connects to the house to fix the leak.
You'll need to seal your dryer vent pipes with adhesive tape if water leaks from the vent. Don't use any adhesive tape, however. You have to buy a special duct tape meant for heating and cooling instruments. This type of tape has a shiny exterior and is meant for sealing aluminum or steel pipes. Get this tape at a specialized hardware store and ask a sales person for help if you aren't sure which type of tape to purchase.
Insulating the vent pipes can also help stop air leaks, thus stopping the moisture inside the vent pipes from turning into water. Use fiberglass insulation to insulate your pipes from cold air. Stuff the entire joint cavity -- the hole between the pipe and the outside world -- with fiberglass insulation. Alternatively, wrap the pipe in the insulation and tape it with heating and cooling duct tape.