The correct order of basicity of the following anions is

Concept:

→ The basicity of an anion refers to its ability to accept a proton. The order of basicity is determined by the stability of the conjugate acid that is formed when a proton is added to the anion.

→ In the given anions, the negative charge is delocalized over the molecule through resonance, making the conjugate acid less stable, and thus the basicity of the anion increases.

Explanation:

→ Nitromethanide:

The nitromethanide anion is stabilized by resonance between the carbon and oxygen atoms.

The conjugate acid formed upon protonation is stabilized by delocalization of the positive charge over the carbon and oxygen atoms, making it less stable than the cyanomethanide conjugate acid. Hence, nitromethanide is less basic than cyanomethanide.

→ Cyanomethanide:

The cyanomethanide anion is stabilized by resonance between the carbon and nitrogen atoms.

Upon protonation, the conjugate acid formed is stabilized by delocalization of the positive charge over the carbon and nitrogen atoms, making it more stable. Hence, it is the most basic anion in the given list.

→ 2,4,6-trinitrophenolate:

The 2,4,6-trinitrophenolate anion is stabilized by resonance between the phenol ring and the nitro groups.

Upon protonation, the conjugate acid formed is stabilized by delocalization of the positive charge over the phenol ring and the nitro groups, making it less stable than the 4-nitrobenzoate conjugate acid. Hence, 2,4,6-trinitrophenolate is the least basic anion in the given list.

→ 4-nitrobenzoate:

The 4-nitrobenzoate anion is stabilized by resonance between the benzene ring and the nitro group.

Upon protonation, the conjugate acid formed is stabilized by delocalization of the positive charge over the benzene ring and the nitro group, making it less stable than the nitromethanide conjugate acid. Hence, 4-nitrobenzoate is less basic than nitromethanide.

Conclusion: 
Therefore, the correct order of basicity is B > A > D > C.