Eric,
Generally speaking, to visualize points such as mine shafts in 3D, there are two things that you need to do.
First, you need to define the height characteristics through the layer properties. You have a couple options available to you, such as the ones below:
Second, since these are points that you want to represent as 3D lines, you will need to extrude the points. You will likely want to use the Base Height type of extrusion and specify a field for the depth of the mine shaft. Because it is negative, you will want to click the green X to make a custom formula. Your formula might be -1*{mine shaft depth} or something similar to that, depending on your data.
Have you tried extrusions and defining the elevation properties? Are you still having issues after working with those two properties?