Moisture inside an air conditioner duct can be caused by insufficient air flow in the air handler. This problem is often caused by a clogged air filter. Air conditioner filters need to be changed regularly; when they become clogged they can impede air flow to the handler and lead to a build up of moisture in the ductwork. Additionally, a clogged air filter will substantially reduce your unit's energy efficiency and longevity.
Moisture accumulating on flexible tubing-style ductwork may indicate the duct insulation has broken down. Over time, the extreme temperature difference and friction from passing air can degrade the duct materials and cause leaking. Flexible ductwork is relatively inexpensive, and replacing the lines is a straightforward weekend project. A sweating duct may be a candidate for a full replacement, with any costs likely being recouped in energy savings.
In the heat of summer, this may seem like an insane suggestion. Raising the thermostat temperature will increase the temperature of the air inside the duct and lead to a reduction in condensation accumulation on the outside of the line. Simply raising the thermostat a few degrees may be all you need to solve your condensation issue.
A duct placed in a crawlspace or basement is much more likely to "sweat" condensation than one in the heat of an attic. It is counter-intuitive, but the increased air temperature leads to an increase in the temperature of the duct, and prevents the surface temperature of the duct from falling below the dew point. This prevents condensation from forming on the duct and causing damage to your home.