Lighting in photography is about ratios, not intensity. You can pretty much expose anything to make it low key but I understand that what you want is to have a darker background than your subject.
The first and easiest way is simply to move the background further. Light does travel at 300,000m/s and will eventually reach your background but it also falls upoff with the square of the distance. So, the further the background, the less bright it will be. What you are looking for itis to adjust the ratio between the light-to-subject and light-to-background distance. The greater that ratio, the darker the background will be compared to the subject.
Another technique is not to light the background. This is useful when creating silhouettes, outlines of objects and patterns of shadows. You can do this by using a directional light, with blinds, reflectors, snoots, etc and place it between the background and the subject. Make sure light goes towards the subject which will be backlit or sidelit, depending on where exactly you place the light source.