Most people look at mold as the problem. The truth is mold is a symptom; a side effect. The real problem is a moisture problem. The best way to prevent toxic black mold (or any other mold for that matter) is to prevent water from infiltrating your home.
Foundation leaks, seepage, condensation or high relative humidity inside the basement can all cause a moist environment for mold to grow. Mold spores exist in the natural environment and decompose organic materials. Therefore, mold needs two things to grow in your house: water and a food source. A food source is any organic material including manmade items made from organic materials i.e. anything made of wood, fabric or fibers. Carpet, insulation, wood framing, traditional sheetrock all become good sources of food for mold.
The easiest element to control in preventing mold growth is water. Keep the relative humidity level at or below 50%. Condition the basement to control the humidity level. The rule of thumb is to treat it like any other space in your house.
However, before you install a dehumidifier make sure that there are no leaks or seepage issues in the basement. Fix any foundation cracks or gaps to prevent future leaks. If there are seepage issues (water infiltrating up through the concrete slab or through the concrete walls) then call a professional basement waterproofer to install a sump pump, interior or exterior waterproofing system. The type of system the contractor recommends will depend on your specific needs and problems with your property. A seepage issue is commonly a sign of water drainage problems on the outside of the house. Check the drains, gutters, downspouts and discharge lines to make sure they are all working properly. Once you have addressed all the potential water problems and completely waterproofed the basement then you can install a dehumidifier. Installing a dehumidifier before all that work is done will only overwork the dehumidifier.
If you have existing mold problems then it is important to call a professional basement waterproofing contractor or mold remediation contractor. Bleach alone does not kill mold. It will clean the surface but it does not remediate mold. The other tricky part about mold is that for every patch of mold you can see there is that much or more hiding behind the walls or under the carpet. A mold remediation contractor will strip the mold completely from the house. If the problem is bad enough that means replacing any finishes that have been damaged beyond repair. Remember, mold is only the sign of a water problem. If you have a serious mold problem, then most likely the damage to the walls or wood framing is going to be from the excessive water.
A professional basement waterproofing contractor can truly help you prevent toxic black mold by giving you a completely dry basement environment where mold will not grow at all.