Correct order of basic strength in gas phase is
(I) `CH_(3)-NH_(2)` (II) `(CH_(3))_(2)NH`
(III) `(CH_(3))_(3)N` (IV) `NH_(3)`

Explanation

`*` The base whose conjugate acid is more stable will be more acidic forming conjugate acid of the given base
`overset(+)NH_(4) , CH_(3)overset(+)NH_(3), (CH_(3))_(2)overset(+)NH_(2), (CH_(3))_(3)overset(+)NH`
Stability order of conjugate acid
`(CH_(3))_(3)overset(+)NH gt (CH_(3))_(2) overset(+)NH_(2) gt CH_(3)overset(+)NH_(3) gt overset(+)NH_(4)`
(due to` +I` effect)
Therefore basic strength
`(CH_(3))_(3)N gt (CH_(3))_(2)NH gt CH_(3)NH_(2) gt NH_(3)`
(vapor phase or gaseous is phase or in Non polar solvent)
In Aqueous solution or in polar solvent
`(CH_(3))_(2)NH gt CH_(3)NH_(2) gt (CH_(3))_(3)N gt NH_(3)`
`*` In aqueous solution, the conjugate acids form H-bonds (intermolecular) with water molecules and stabilise them selves conjugat acid of `1^(@)` amine which has largest no. of H-atoms form maximum H-bond with water and is most stable. Consequently `1^(@)` amine is most basic.
`*` Due to steric effect `1^(@)` amine is considered more basic as compared to `3^(@)` amine as lone pair is hindered by three alkyl group and less available for `H^(+)`.
Considering the combined effect of the three (Inductive, solvation and steric effect) we can conclude that
`2^(@) gt 1^(@) gt 3^(@) gt NH_(3)`
`*` Aromatic amines are least basic as their lone pair is in conjugation and less avaibable for protonation.