Might not be something that can be done automatically with modern processes (that board looks like it might have been a rather old hand drawn design). But in your PCB CAD software you could try copying the wire traces and pasting the copy in to the silkscreen layer (most PCB fabs won't do multi-coloured soldermask), or if the software (like Eagle)won't let objects definedin copepr layers be copied to other layer types then trace over your traces by hand in the silkscreen layer, or export an image of the PCB's copper layer, process it to some appropriate format (dxf maybe) and reimport it in to the silkscreen layer.
Actually having solder mask only cover copper traces would be a bad idea, as the alignment won't always be perfect and then you'll have slight slivers of exposed copper to short things to or make shorting while soldering easier, unless you had the solder mask cover the copper PLUS an offset distances around all the copper.