Step 1: Note why the gas phase order does not carry over to water.
In the gas phase, more alkyl groups simply push more electron density onto nitrogen, so basicity would run $3^\circ \gt 2^\circ \gt 1^\circ$. In water, though, two more effects come into play once the amine grabs a proton, how well the resulting cation is stabilized by water molecules through solvation, and how much the bulky alkyl groups get in the way of both the proton and the water molecules through steric hindrance.
Step 2: Weigh the three effects together.
The $+I$ effect alone favours the tertiary amine, solvation alone favours the primary amine since it has more $N-H$ bonds free for hydrogen bonding, and steric bulk works against the tertiary amine the most. For the ethyl series, these effects settle into a balance where the secondary amine comes out on top.
Step 3: Rank the three amines.
$(C_2H_5)_2NH$ gets a good boost from the $+I$ effect of two ethyl groups without suffering the crowding a third group would bring, so it is the strongest base; $(C_2H_5)_3N$ comes next since its steric hindrance is severe but its inductive push is still large; $C_2H_5NH_2$, being the least alkylated, is the weakest base of the three.
Step 4: State the final order.
\[ \boxed{(C_2H_5)_2NH \gt (C_2H_5)_3N \gt C_2H_5NH_2} \]