Removing the existing basement mold is not enough to keep it out of your home. The dead spores have the ability to survive and within a few days or weeks it will return. Instead you have to go through the process of kill the mold and then learning how to prevent it.
There are several techniques that can be used when trying to kill mold. The first is to purchase mold killers from the store. While these are effective they are made up of harsh chemicals that could be dangerous when not used properly. For people who are uncomfortable with this there are other, safer methods that you can use.
Lemon juice, vinegar, and other types of strong acids are effective at killing basement mold. A solution made with tea tree oil and grapefruit seed extract with warm water is also an effective way to kill it. The one product you should never use is bleach. This will stain the objects that it comes into contact with unless they are white.
When killing the mold spray the vinegar or one of the other solutions onto the affected area. Let it soak for a few moments before using an old rag to remove it. During this time that ventilation system should be turned off and all the vents covered with plastic. Leave it off for awhile to allow the room to heat up.
Heat is effective at killing mold because it will remove moisture from the area and keep it from coming back anytime soon. Any curtains, clothes, or other materials that had mold on it should either be thrown away or washed with a cup of vinegar and dried with high heat.
Along with high heat you should place an air purifier in the basement that will keep the humidity low. You can also accomplish this by opening windows if there are any down there. Any drywall or fixtures that were contaminated with mold should be removed and replaced right away.