My air conditioner specifically has a 'dry' setting, which removes moisture from the air better than the regular setting (although air conditioners in general do dehumidify to some extent; you'll notice they drop water outside). Reducing humidity can reduce mold, as others have said—however, humidity is probably not the most important thing you should worry about. I would be more concerned about a lack of air circulation.
Old or dirty air conditioners can introduce dust and things that might be unpleasant for some people to breathe. The particles could potentially include mold spores. However, the dehumidification and the air-flow will probably prevent those microbes from growing while the air conditioner is going (although after it stops, it could potentially grow faster than before the air conditioner was used—but I wouldn't count on it).
Although mold does appreciate warm temperatures for growth, it is my experience that heat combined with air circulation does cause a drier environment than cold combined with air circulation (notwithstanding air conditioners do dehumidify). I use a warm room and a box fan to dry seeds quickly, without mold (for gardening), and I turn the air conditioner on dry for a bit when it gets too humid (although with the fan going, I haven't found extra mold during humid times, but I do feel better having drier air, and not just dry surfaces). You'll notice that food driers blow warm air (the same principles work for drying other stuff). In my experience, mold doesn't grow easily if there is air circulation in the room (without regard to temperature), however, but things dry faster if it's hot.
I would personally recommend having fans going (like box fans; I use a box fan to keep black mold from growing on the walls in my room; I have one going almost all of the time, and I haven't had black mold there since; it's been years). Air circulation helps to keep things dry. I also use an ionizer to help prevent mold (supposedly, the negative ions can kill microbes in the air), and I personally think it works based on my use of it (but I haven't done a lot of experiments there, and I don't mean to be making any claims there; I had already had success with the fans before getting the ionizer).
Disclaimer: I live in a dry climate—so my humid times may not be nearly as humid as yours.
I should probably point out that for some reason I don't understand, central air doesn't seem to prevent mold as well as a box fan (in fact, I can see how someone might think that would promote mold, although I don't have any data to support that notion beyond how food got moldy pretty easily in the house with central air). Perhaps central air failed me with regard to mold because it spread microbes from every room in the house to every other room (just a guess), and the vents were old. I don't know that central air helps to dehumidify much, but regular air conditioners you put in the window can do so quite a bit.
While box fans may not exactly dehumidify, they do help to keep the surfaces in the room dry.
