Safest Methylating Agent
What you are looking for does not — to the best of my knowledge — exist and likely won’t ever exist.

Methylating agents are usually classified as toxic as they are usually able to penetrate cells and nuclei and methylate DNA bases — although they will also methylate most nucleophiles on the way there. This can lead to all sorts of things will the shadow of cancer overarching it all. Off the top of my head there are a few methylating agents that might not penetrate the skin such as methyl Meerwein salt. These, however, are so reactive that they just react before they reach the skin.

Secondly, methylating agents will likely never work in water. Water itself is a nucleophile and a (Brønsted) acid. If you use an electrophilic methylating agent such as methyl iodide or methyl Meerwein salt, water is able to nucleophilically attack these to generate methanol and then dimethyl ether, thereby deactivating your methylating agent. If you choose to use a nucleophilic methylating agent such as methyllithium instead, water will supply a proton to liberate methane (and lithium hydroxide in the case of methyllithium) instead. So practically no methylating agent will survive in aquaeous solution.

Finally, your proposed reaction pathway is skewed from the beginning: you suggest using a deprotonated amine in water as a reactant. However, the $\mathrm{p}K_\mathrm{a}$ of amines is typically around $30$, depending on the substituents. ($\mathrm{p}K_\mathrm{a}(\ce{LDA}) \approx 35; \mathrm{p}K_\mathrm{a}(\ce{LiHMDS}) \approx 25$; source: Evans $\mathrm{p}K_\mathrm{a}$ table.) On the other hand, the $\mathrm{p}K_\mathrm{a}$ of water is $14$ or $15.7$, depending on the source you consult. This is a difference of $10$ to $20$ logarithmic units depending on the type of amine — meaning that amines are practically never deprotonated in aquaeous solution.[1]

If you mistyped and are using a deprotonated amide, things won’t get that much better ($\mathrm{p}K_\mathrm{a}(\ce{RCONH2}) \approx 25$).

Therefore, you will likely have to give up your plan and go for a more traditional methylation method.



[1]: Note you might argue that the free base, $\ce{R3N}$ is enough, so all you want to do is lower the $\mathrm{pH}$ of water to below $\approx 10$ so that the protonated form $\ce{R3NH+}$ is neglegible. However, at these $\mathrm{pH}$ values, water is also deprotonated to a non-neglegible amount and a hydroxide ion is a better nucleophile than a free amine base.