When dealing with any style of mold the first thing to do would be to throw out the object(s) that is supporting the mold; in this case the rug. Letting anything sit and decompose because of mold isn’t healthy. The mold that will grow on that rug adds many pounds of allergens to the air each day.
Second thing to do would be to scrub down the floor with a mop using a 50/50 mix of bleach to water and let dry.
Whenever you work around mold, whether you’re sensitive to mold or not, make sure to wear proper eye, hand and clothing protection in order to not let your skin or eyes become irritated. If your skin does come into direct contact with mold, make sure to wash your skin with soap and warm water for at least 30 seconds.
Thirdly, this is the time to have a very thorough basement inspection. Ask yourself: where is the water/moisture coming into the basement? Am I running a dehumidifier to help filter the air and regulate the RH (relative humidity) in the basement? What other organic objects (boxes, bookshelves, wood, clothing, paper, etc.) is down here that might be feeding the mold?
By removing all the mold food you can easily gain control back of your basement. But don’t neglect the basement because of one small rug being moldy. This rug was a definite sign of issues in your basement that needed to be handled.
Contact your local basement-waterproofing expert in order to fully control any and all moisture in your basement. Controlling the moisture is a sure way to limit and even eliminate the mold content of your basement.