Why are Group 11 elements unreactive?
The first step toward answering this question is understanding the extent to which the statement "Group 11 metals are unreactive" is or isn't true.  

As stated in How do plants respond to copper deficiency? Plant Signaling and Behavior vol. 3, pages 231-232:


  The transition metal copper is essential for all organisms.


It is not just that copper is present in every living thing.  Copper is reacting in every living thing. Copper ions cycle between the +1 and +2 oxidation states in Plastocyanin,  Copper/zinc superoxide dismutase and cytochrome oxidase. So as you are saying copper is unreactive, it as reacting in cytochrome oxidase in every mitochondrion of your body, and in Cu/Zn superoxide dismutase in the cytosol of cells through your body. 

One way to quantify the reactivity of metal ions in through reduction potential, here listed for reduction of the ion to the zero oxidation state metal.  (limiting to d-block metals at least as unreactive as Cu)

\begin{array}{c @{:} c} \\
\color{blue}{\text{Ion}} &amp; \color{blue}{\text{Reduction potential}} \\\hline
\mathbf{\color{red}{Cu^{2+}}} &amp; 0.3419  \\\hline
\ce{Tc^2+} &amp; 0.400   \\\hline
\ce{Ru^2+} &amp; 0.455  \\\hline
\ce{Cu+} &amp; 0.521 \\ \hline
\ce{Rh+} &amp; 0.600  \\
\ce{Rh^2+} &amp; 0.600  \\
\ce{Rh^3+} &amp; 0.758  \\ \hline
\mathbf{\color{red}{Ag^+}} &amp; 0.7996   \\\hline
\ce{Os^2+} &amp; 0.85  \\\hline
\ce{Hg^2+} &amp; 0.851  \\\hline
\ce{Pd^2+} &amp; 0.951   \\\hline
\ce{Ir^3+} &amp; 1.156  \\\hline
\ce{Pt^2+} &amp; 1.18   \\\hline
\mathbf{\color{red}{Au^{3+}}} &amp; 1.498  \\
\ce{Au+} &amp; 1.692  \\
\end{array}

Looking at this list, we see that there are a substantial number of d-block metals that are less reactive than copper.  A few are less reactive than silver, and gold is the least reactive on the basis standard reduction potential. 

The overall trend in d-block metals, is down and to the right in the block is less reactive. The exception is that, for a given row, the group 12 metal is more reactive than the group 11 metal, but due to the formation of a 2+ ion, not a 1+ ion.  

Next we should look at ionization energies of d-block metals to see how much of the reduction potential corresponds to properties of an individual atom, versus bulk or solvent effects.   

Looking at first-ionization energies, for each row of the d-block, the highest ionization energy is always the group 12 element (Zn, Cd, Hg). This corresponds well with the fact the Zn, Cd and Hg do not form +1 ions.  The group 11 elements have significantly lower first-ionization energies, and do form +1 ions.  

Then looking at second-ionization energies, group 11 has the highest second-ionization energies for every row of the d-block!

Much of the reactive/unreactivity of d-block is explained by the increase of nuclear charge to the right and down the d-block.  The complete d-shell of neutral group 11 atoms shields the valence s electron well enough that it may be removed to react.  However, to lose 2 electrons, the group 11 elements would need to lose a d-electron from a completed d-shell, which is difficult.  For group 12, two s valance electrons can be lost, and the complete d-shell maintained.