[Kovner]

What can the button have here that we'd want to value bet against? a 6? It's more likely he's calling down with an 8 (some passives like to call down with TP on suited flop), or more likely a club (he called two bets cold, and then another bet on the turn - looks like a draw).

If he's calling down with an 8, we'll lose the least amount when we check and he checks behind (worst case scenario is he bets, we call and lose the same amount as betting and getting called).
If he's drawing to a club, he'll usually fold to a bet and sometimes bet when checked to (even though he's passive, this is more likely then him calling with high-card). Worst case scenario when he has high club is he checks behind and we gain the same amount (zero bets) as betting and him folding.

Checking, therefore, costs the least when we're behind and makes the most when we're ahead.