I think you do have condensation and you have always had condensation. Without the "waterproofing" material, the blocks absorbed the moisture, as the bare areas are doing now. The "waterproofing" is preventing the moisture from being absorbed so it beads up on the surface.

Waterproofing of the basement block walls should be done on the outside, and since you don't seem to have a problem where the waterproofing is missing, I'd guess that it has been done and is still in good shape. Properly waterproofing the inside of the blocks will help them last longer.

The only way to stop the condensation is to reduce the humidity in the air in the basement. If you have a laundry room in the basement, chances are the relative humidity in the basement is pretty high, especially on laundry days. Since part of the basement walls are exposed to the outdoor temperature and the basement is not heated, then the basement walls are cold enough to be below the dew point for the relative humidity in the room, so condensation will form.

A dehumidifier would be your best bet here. Even then, you may still see some condensation on the walls.