Vice President Joe Biden's son and a close friend of Secretary of State John Kerry's stepson have joined the board of a Ukrainian gas producer controlled by a former top security and energy official for deposed President Viktor Yanukovych.

The move has attracted attention given Messrs. Biden's and Kerry's public roles in diplomacy toward Ukraine, where the U.S. expressed support for pro-Western demonstrators who toppled Mr. Yanukovych's Kremlin-backed government in February. The uprising provoked a pro-Russia backlash that...