Effect of sodium hydroxide solution on copper plate
Regarding the copper metal itself, the only down side I see of using a $\ce{NaOH}$ is that it is unlikely to remove some of the oxides that would be covering any exposed copper.  The initial etching process would have mainly produced water-soluble $\ce{CuCl2}$, likely with a bit of insoluble $\ce{CuO}$.  $\ce{CuO}$, $\ce{CuCO3}$ and some $\ce{Cu(OH)2}$ would have likely formed on any unprotected copper surfaces over time.  $\ce{NaOH}$ could remove $\ce{Cu(OH)2}$ and $\ce{CuO3}$, but not $\ce{CuO}$.  It also should be unreactive toward the copper plate itself.

A common method of putting a good shine on tarnished copper is with weak organic acids like citric acid.  Lemon juice and table salt make a favorite home copper cleaning solution.  Again, this is only with regards to the cleaning oxides from the copper metal itself, not the ink residues coating it.  

My bottom line advise is to do what I’ve seen in your last comment that you plan to do; test a portion of the bottom side of the plate with the $\ce{NaOH}$ solution to verify that this solution cleans the ink residues without harming the copper.  I think you can do this with confidence that the copper metal will be unharmed.  If the ink residues are removed and you want to remove any remaining oxidation to create a fresh, shiney copper plate, you could dissolve just a bit of salt into lemmon juice and wipe the plate with this solution until shiny!