First we have to understand what affects basic nature of a compound.
- Tendency to donate lone pair
- Electronic effects— Inductive, resonance etc
- Factors stabilizing conjugate acid (or acidity of conjugate acid
- Nature of solvent
Tendency to donate lone pair is affected by the acidity of conjugate acid and electronic effects.
Grab a pen and paper and draw conjugate acids of
And you know that electron withdrawing inductive (–I) effect increases acidity and electron withdrawing resonance (–M or –R) also increases acidity.
In conjugate acid of
You will find there is only –I inductive effect and there is no –M effect of $-NO_2$ group because of SIR (In SIR - STERIC INHIBITION OF RESONANCE - the nitro $-NO_2$ group shows steric hindrance to $-NH_3^+$ group and the p–orbital of Nitrogen (in $-NH_3^+$ group) moves out of plane of p–orbital of Carbon (in benzene ring) which inhibits resonance or does not allow resonance to happen). So in conjugate acid of $2-nitroanliline$ there is only –I effect which increases acidity.
You will find that there is no –M effect obviously!! But there is –I effect of nitro ($-NO_2$) group which increase the acidity of conjugate acid of $3-nitroanliline$ but it is less than what we have seen in $2-nitroanliline$ (due to distance factors). Therefore conjugate acid of $2-nitroanliline$ is more acidic than conjugate acid of $3-nitroanliline$
You will find there are both –M and –I effect which increase acidity of conjugate acid of $4-nitroanliline$. So conjugate acid of $4-nitroanliline$ is more acidic than conjugate acid of $3-nitroanliline$ or $2-nitroanliline$
Order of acidity of conjugate acids
$$conj.\ acid\ of\ 3-nitroanliline<conj.\ acid\ of\ 2-nitroanliline<conj.\ acid\ of\ 4-nitroanliline$$
Since the order of basicity of 'orginal' compounds is just opposite of order of acidity of conjugate acids, so answer to your question is as follows