Change the voice: Don’t mistrust the sincere.

Let the sincere not be mistrusted.

Let the sincere not being mistrusted.

Let the sincere not to be mistrusted.

None of these

Negative imperative in passive voice uses: "Let + object + not be + past participle."

"Don’t mistrust" becomes "Let ... not be mistrusted."

Hence, "Let the sincere not be mistrusted" is correct.