The correct order of increasing basic nature for the bases ${\text{NH}}_{\text{3}}{\text{,\hspace{0.17em}CH}}_{\text{3}}{\text{NH}}_{\text{2}}$ and  ${\left({\text{CH}}_{\text{3}}\right)}_{\text{2}}\text{NH}$ is:

The alkyl groups donate electrons to the nitrogen atom by +I effect. Among the amines, secondary amines are more basic than primary amines. So, the correct order of increasing basic nature of bases in aqueous solution is NH₃<CH₃NH₂<(CH₃)₂NH.