Basic strength in aqueous medium depends on two main factors:

• $+I$ effect of alkyl groups, which increases electron density on nitrogen and tends to increase basicity.
• Solvation of the conjugate acid in water, which stabilizes the protonated amine. Better solvation means greater basic strength in aqueous medium.

For methyl-substituted amines:

• $\mathrm{MeNH_2}$ (primary amine)
• $\mathrm{Me_2NH}$ (secondary amine)
• $\mathrm{Me_3N}$ (tertiary amine)

Although tertiary amine has the strongest $+I$ effect, its protonated form is poorly solvated due to steric hindrance.

In aqueous solution, the usual order of basic strength of aliphatic amines is: $\mathrm{2^\circ > 1^\circ > 3^\circ > NH_3}$

Applying this to methyl amines: $\mathrm{Me_2NH > MeNH_2 > Me_3N > NH_3}$

Now check the options:

• A: $\mathrm{NH_3 > Me_3N > MeNH_2 > Me_2NH}$ ❌ incorrect
• B: $\mathrm{Me_3N > Me_2NH > MeNH_2 > NH_3}$ ❌ incorrect
• C: $\mathrm{Me_2NH > MeNH_2 > Me_3N > NH_3}$ ✅ correct
• D: $\mathrm{Me_2NH > Me_3N > MeNH_2 > NH_3}$ ❌ incorrect

Therefore, the correct order is: $\boxed{\mathrm{Me_2NH > MeNH_2 > Me_3N > NH_3}}$