The correct stability order of the following diazonium salts is

The stability of benzenediazonium salts is governed by the electronic effects of the substituents present on the benzene ring.
The diazonium group $-N_2^+$ is strongly electron-withdrawing. Electron-donating groups (EDG) stabilize the diazonium cation by dispersing the positive charge through resonance or inductive effects, whereas electron-withdrawing groups (EWG) destabilize it by increasing the positive charge density.
In (A), the $-OCH_3$ group at the para position is a strong electron-donating group via the $+M$ effect, which significantly stabilizes the diazonium ion.
In (C), there is no substituent (hydrogen), providing a baseline stability.
In (D), the $-CN$ group is an electron-withdrawing group via $-M$ and $-I$ effects, which destabilizes the cation.
In (B), the $-NO_2$ group is a very strong electron-withdrawing group via $-M$ and $-I$ effects, even stronger than $-CN$, leading to the least stability.
Comparing the effects: $+M$ of $-OCH_3$ (A) > $H$ (C) > $-M$ of $-CN$ (D) > $-M$ of $-NO_2$ (B).
Therefore, the correct stability order is $A > C > D > B$.