On a paired board that favors the defender, we should check our high-card hands to protect our range and realize equity.
Checking is the preferred play here. While we have the range advantage, this paired board hits the BB's flatting range quite well, and our specific hand has enough showdown value to check-call.
Note: C-betting here is slightly over-aggressive; checking allows us to realize equity with our Ace-high and protects our checking range on a board that favors the defender.
Checking is pure. The second six on the turn further connects the board, and since we are now playing the board's two pair, our hand functions as a mid-strength bluff catcher.
Checking is the standard play to reach showdown. Our Ace-high kicker is strong enough to win against BB's missed straight draws or lower high-card hands that check back.