For a typical basement room, with concrete walls and a low ceiling, your best strategy is to use as much acoustical absorption as is practical.
The idea is to make the room as dead as possible, because the room itself will probably not have naturally pleasant reverberant properties. The vocal recordings you make in such a room will be very dry, but you can liven them with artificial reverb.
The most popular material for this is Owens-Corning 703 (rigid fiberglass) or equivalent, such as Rockwool (generically "mineral wool") or Roxul. Your building supply store or an insulation supplier can suggest alternative brands, although it's likely you'll have to special-order them. There are also online sources, but you'll have to pay shipping so the savings aren't huge. 703 comes in 1", 2" and 4" thicknesses. Get whatever's available and layer them if necessary to get the desired thickness, which will be 3-4 inches, thicker in the corners if possible.
Build frames, preferably tall enough to go from floor to ceiling, and fill them with 4 inches of fiberglass. Fill the room's corners with extra-thick fiberglass insulation. Hang a frame from the ceiling by chains, as thick and as far from the ceiling as is practical without bumping your head. Staple burlap or other porous fabric over their faces to contain the fiberglass and make them look nicer.
The panels can be attached to the wall or free-standing. I prefer movable panels so you can reconfigure them as needed, e.g. making room for more than one singer. Whether attached or free-standing, you want them at least 4 inches from the wall if possible.
If it's an unfinished basement with exposed floor joists, fill the space between the joists with regular pink fluffy insulation.