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How to best draw an additional box boder at {x, y, 0} as indicated by the red arrow below? PlotRange is Automatic (do not assume it is {{0, 100}, {0, 100}, {-50, 50}} always).

points = RandomReal[100, {100, 3}];

points[[All, 3]] = points[[All, 3]] - 50;

ListPointPlot3D[points, BoxRatios -> 1]

The only idea I have is extracting PlotRange with AbsoluteOptions and drawing the box border manually. That may be too difficult for me. I use Mathematica 9.

You can use InfinitePlane without having to get the PlotRange of input plot:

Show[ListPointPlot3D[points, BoxRatios -> 1],

 Graphics3D[{Opacity[0], EdgeForm[{Blue, Thick}], 

   InfinitePlane[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}}]}]]

Update: An alternative that also works in version 9 is to use FaceGrids:

facegrids = {#, {{}, {0}}} & /@ Join[#, -#] &@Most[IdentityMatrix[3]];

ListPointPlot3D[points, BoxRatios -> 1, FaceGrids -> facegrids, 

 FaceGridsStyle -> Directive[Thick, Blue]]

Update 2: You can also use PlotRange to extract the plot range of a plot object and use it with Cuboid:

lpp = ListPointPlot3D[points, BoxRatios -> 1]; 

rectangle = Transpose[Append[PlotRange[lpp][[;; 2]], {0, 0}]];

Show[lpp, Graphics3D[{Opacity[0], EdgeForm[{Thick, Blue}], Cuboid @@ rectangle}]]

You can use the useful-but-undocumented function Charting`get3DPlotRange to find the plot range, and use that to make your box:

points = RandomReal[{-50, 50}, {100, 3}];

plot = ListPointPlot3D[points, BoxRatios -> 1];

{x, y, z} = Charting`get3DPlotRange @ plot;

Show[plot,

    Graphics3D[

        {

            EdgeForm @ Blue,

            FaceForm @ Opacity @ 0.05, (* set to 0 for transparent *)

            Cuboid @@ Thread[{x, y, {0, 0}}]

        }

    ]

]