It's at the start of each turn, including opponent turns (assuming the designers are consistent in their wording).
We can use the text on some other cards to work this out. First, we can look at cards that specify an action on the player's turn, which are very specific. This includes cards such as Herbivore and Metamorph (emphasis mine):
Herbivore: Gain 1[Star] on your turn if you don't damage anyone.
Metamorph: At the end of your turn you can discard any keep cards you have to receive the [Energy] they were purchased for.
We can then look at other cards that use the terminology of "each", which are abilities you would only use on an opponent's turn. Two such cards are Super Jump and Psychic Probe (emphasis mine):
Super Jump: Once each turn you may spend 1[Energy] to negate 1 damage you are receiving.
Psychic Probe: You can reroll a die of each other monster once each turn. If the reroll is [Heart] discard this card.
In the first case, you'd negate damage during an opponent's turn, and in the second you would choose to reroll a die on an opponent's turn. So the phrasing "each turn" in Monster Batteries indicates that it occurs at the beginning of each turn around the table, whether it is yours or not.