Do the Open Aerial Map Landsat tiles not work for you? You do need to add their wms address to the JOSM wms plugin to make it work, but this procedure is described in their wiki. Have you noticed that you can also save the downloaded tiles in JOSM and use them later from your own machine.
I have not yet found a way to use the tiles from NASAs (non-volatile-wms) Landsat site directly on JOSM, though. This should be possible somehow, shouldn’t it.