More be the oxidation state of central atom
(i.e. $\mathrm{Cl}$-atom), more is the acidic nature of the species and lesser will be the basic strength.
As, oxidation number of :
$\begin{aligned} \mathrm{Cl} \text { in } \mathrm{ClO}_2^{-} \rightarrow x-4 & =-1 \\ x & =+3 \\ \mathrm{Cl} \mathrm{in} \mathrm{ClO}_3^{-} \rightarrow x-6 & =-1 \\ x & =+5 \\ \mathrm{C} \mathrm{lin} \mathrm{ClO}_4^{-} \rightarrow x-8 & =-1 \\ x & =+7\end{aligned}$
Thus, order of acidic nature is
$\mathrm{ClO}_4^{-}>\mathrm{ClO}_3^{-}>\mathrm{ClO}_2^{-}$
and order for basic strength is
$\mathrm{ClO}_2^{-}>\mathrm{ClO}_3^{-}>\mathrm{ClO}_4^{-}$.
Alternate method
The more be the oxygen atom in the given species, the more is the dispersion of electrons from Cl-atoms, which decreases the basic strength. $\because \mathrm{ClO}_2^{-}$has only two oxygen atoms while $\mathrm{ClO}_4^{-}$has four oxygen atoms. Thus, order of basic strength is
$\mathrm{ClO}_2^{-}>\mathrm{ClO}_3^{-}>\mathrm{ClO}_4^{-}$
Hence, option (a) is the correct answer.