Among the following compounds, the increasing order of their basic strength is:

Basicity of the molecule depends on the availability of the lone pair on the Nitrogen atom.

In (II) the lone pair is a part of the aromatic 6$$\pi$$ electron system. Thus the lone pair is extremely unavailable making (II) the least basic compound.

In (I), the lone pair is involved in resonance with the benzene ring, making it less available.

In (III) and (IV), the lone pair is localized and available for protonation. (IV) is more basic due to lower steric hindrance at the lone pair, making it easier to protonate.

Final basicity order: (IV) > (III) > (I) > (II)