When direct resizing causes visible artifacts (aliasing or moiré patterns), it often helps if the resize is done in several steps. For instance a 50% resize is equivalent to three consecutive 80% resizes (plus one 97.66%, if you want to be exact): 0.5 = 0.8*0.8*0.8*0.9766
This is demonstrated, I hope, with the attached files
021_large, 238_large -> original size, cleaned, cropped, etc.
021_aliased, 238_moire -> direct 50% resize
021_better, 238_better -> multi-step resize, plus levels to remove some of the blurriness (use the burn layer trick again to see that).
I'll leave the color reductions and lossless PNG optimizations for others
