Air conditioners do dehumidify the air, but that is not their main purpose. The dehumidifying aspect of a central air conditioner is something of a side effect or byproduct of the cooling cooling. A central air conditioner in average humidity could put out a gallon or more of water per hour, but a dehumidifier could remove two or three times that amount of water vapor from the air.
Cooling indoor air is significantly more energy expensive than dehumidifying the air alone. An air conditioner has numerous components, including air handlers and condensers, that a dehumidifier does not possess. These components increase the electrical usage of an air conditioner over a dehumidifier. As such, running an air conditioner as a dehumidifier is much more expensive than running a standalone dehumidifying unit.
Most humid basements are problematic for a homeowner chiefly because of their odor. A central air conditioner takes in air from a few intake points, and blows back cold air through registers in every room. If you rely on the central air conditioner to dehumidify your basement, realize that you are going to have the same smell you have in your basement throughout most of the house.
Many modern thermostats turn the air conditioner on and off based on two readings. The first is temperature; when the temperature falls below a certain level, the air conditioner shuts off. The second reading is humidity; an air conditioner continues to run until the indoor humidity falls below a certain level. Opening up your basement to the air conditioner means this humidity reading is going to take significantly longer to be brought down, which means the air conditioner is going to run much longer.