How to add a common color legend to a grid of density plots?

As Cormullion pointed out, you will have to create the legend by hand as it would be difficult to have a legend created automatically by GraphicsGrid. However, the simplest method of attaching a legend is to use Legended directly. The structure of Legended when used like this is

Legended[ Graph | Graphics | Graphics3D, legend ]

Then, in your case, you use

Legended[
  GraphicsGrid[Partition[Table[
    DensityPlot[Sin[i x] Sin[y/i], {x, -4, 4}, {y, -3, 3},
      ColorFunction -> "LakeColors"],
    {i, 1, 4}], 2]
  ],
  BarLegend[{"LakeColors", {-1, 1}}, LegendLayout -> "Column"]
]

Or, if you would like to place it below

Legended[
  GraphicsGrid[Partition[Table[
    DensityPlot[Sin[i x] Sin[y/i], {x, -4, 4}, {y, -3, 3},
      ColorFunction -> "LakeColors"],
    {i, 1, 4}], 2]
  ],
  Placed[
    BarLegend[{"LakeColors", {-1, 1}}, LegendLayout -> "Row"],
    Below
  ]
]

When generated automatically using PlotLegends, Placed is inserted automatically, so you can use it to position the legend here.

Best I could do. GraphicsGrid can't read my mind at all...

legend = BarLegend[{"LakeColors", {-1, 1}}, LegendLayout -> "Row"];

plots = Table[
   DensityPlot[Sin[i x] Sin[y/i], {x, -4, 4}, {y, -3, 3}], {i, 1, 
    4}];

Column[{
  GraphicsGrid[{
    {plots[[1]], plots[[2]]},
    {plots[[3]], plots[[4]]}}, ImageSize -> 300],
  legend}, Center]

I don't have 9 on this computer but how about:

Labeled[
 grid = GraphicsGrid[
   Partition[
    Table[DensityPlot[Sin[i x] Sin[y/i], {x, -4, 4}, {y, -3, 3}], {i, 
      1, 4}], 2]],

 DensityPlot[y, {y, -1, 1}, {x, 0, 1}, AspectRatio -> 0.07, 
  ColorFunction -> "LakeColors", 
  FrameTicks -> {{None, None}, {Automatic, None}}, FrameStyle -> GrayLevel[0.6],      
  FrameTicksStyle -> GrayLevel[0.],
  ImageSize -> First@AbsoluteCurrentValue[grid, ImageSize], 
  Mesh -> False, PlotRangePadding -> 0]
 ]

To get outside ticks on the legend you could either enter a tick list manually, or make a tick function, or do a replace(all) on the DensityPlot.

 ticks = Join[
   Table[{i, "", {0, 0.01}, GrayLevel[0.6]}, {i, -1, 1, 0.1}], {{-1., 
     "-1.0", {0, 0.015}}, {-0.5, "-0.5", {0, 0.015}}, {0, 
     "0", {0, 0.015}}, {0.5, "0.5", {0, 0.015}}, {1., 
     "1.0", {0, 0.015}}}];

These ticks, when compared with the built in label, look a little high. So we can nudge them lower. Highlight the value and press "Control" + "Down arrow" or go to the menu Insert > Typesetting > Nudge Down. This wraps adjustment boxes around the tick label.

After nudging the ticks down you get this: