I know some will suggest indoor/outdoor carpeting. This is a good choice. If you want an indoor only carpet, you may want to get a dehumidifier as well.
I would recommend throw rugs 8x10 or 5x8 versus wall-to-wall. If the rug(s) get wet, you will need to hang it/them outside to dry. A wet rug will remain wet for months in a humid basement and grow mold and mildew.
I say throw rugs & dehumidifier.
They make rubber pads to protect the rugs that look like a rubber links with holes. I would recommend this type of pad.
If it were me, I would use the cheap machine made orientals that run about $50 for a 5x8.