The turns of a solenoid, designed to provide a given magnetic flux...

The cross-sectional area of the space to be filled is fixed, whilst that of the wire varies as $d^2$. Thus, $n \propto d^{-2}$. The resistance of one turn is inversely proportional to the cross-sectional area of the wire, i.e. varies as $d^{-2}$ and hence the resistance per unit length of the solenoid is $R \propto n d^{-2} \propto d^{-4}$. The flux density $B$ is $\propto n I$ and therefore the required current $I \propto n^{-1} \propto d^2$. The heat dissipated per unit length is $R I^2$, which $\propto d^{-4}\left(d^2\right)^2$, i.e. independent of $d$. Thus, within limits, it does not matter what diameter wire is chosen so far as the heating effect is concerned.