If your pipes in the basement have been sweating and you now have a mold problem, you'll want to call a mold remediation company right away. The mold will spread quickly, so it's important to get the situation taken care of before other areas in the house are affected.
All people will react to mold differently, but some will have allergic reactions and respiratory illnesses. It can also cause problems for people with asthma. The company will first have to dry the basement out, and they will use large heaters and fans. Once the area is dry, they will start the mold removal process.
The mold removal company will come in suck the moisture and the mold out of the affected areas, and they will apply a coating that will work to kill the mold living around the space. This will kill the mold that you can see, and the microscopic mold spores that are lingering in the drywall, wood and other areas around the home.
You should consider putting a waterproofing agent around the floor and foundation around the basement to prevent future moisture damage. A sealant is going to prevent moisture from weakening the foundation, and it can be applied to the walls as well.
Add insulation around the pipes to prevent future condensation from occurring. This is going to help prevent the pipes from getting cold enough to cause condensation when hot water goes through, and it's going to stop condensation from dripping around the basement. Examine the pipes to make sure there isn't a loose fixture or any damage, which could cause water to leak.
Have a dehumidifier set up in the basement so you don't have to worry about a lot of moisture in the air again. The machine will have a humidistat that determines the amount of moisture in the air, and will turn on when moisture is clear problem. This will also prevent you from having a damp feeling and musty smell in your basement.
The mold removal company can look to see if the mold got into the ventilation system, and if you need to have your ducts cleaned out. You don't want to ignore the problem if you see greenish or white spots spreading across the walls and around the pipes, because it can get into the insulation around your home and in other areas. Mold remediation companies are the experts, so let them fix the problem.