The "sweating" is condensation. When warm, humid air contacts a cold surface such as your air-conditioning ductwork, the water vapor in the air will condense on the surface of the cold object.
This occurs because the air that contacts the cold surface is cooled sufficiently to return the water vapor it holds to the liquid phase.
Other Examples of Condensation
Another familar example of condensation is the water that forms on the outside and bottom of a cold glass of ice water. The same principle is in play: The cold water cools the surface of the glass, and the cold glass cools the surrounding air.
Moisture in the cooled air reverts to the liquid phase on the surface of the glass.
Solution 1: Dehumidifier
The simplest solution to the sweating ductwork problem is to install a basement dehumidifier. This small, electric device pulls moisture from the air, lowering the humidity in the basement.
If less water is in the air, less will condense on the ductwork, and the problem is solved. However, dehumidifiers do not work well in temperatures below 65 degrees Fahrenheit, so you may want to invest in one rated for cold weather if you expect to run one in your basement around the clock.
Solution 2: Insulate Ductwork
A more permanent solution is to insulate the ductwork. Properly insulated ductwork will have no cold surfaces exposed to the humid air, and condensation will not occur.
At least as importantly, more of the cold air generated by your air conditioner will go toward cooling your home, rather than the basement. If you use the ductwork to carry heated air in winter, the insulation also will cut heat loss to the basement.
Installing better insulation on the ductwork involves some initial cost, but it solves the condensation problem and may reduce your energy bill.