The detail from Rudyard Kipling’s poem "If" that best supports the theme of the importance of honesty is:
"If you can trust yourself when all men doubt you."
This line emphasizes the significance of self-trust and integrity, suggesting that honesty to oneself is crucial, even in the face of doubt from others. It highlights the idea that maintaining one's own truth and values, irrespective of external opinions, is a vital quality in living an honest life.