I would bet moer on the flop, for starters. Thats quite the draw-heavy board and a thinking opponent would think you would bet more to protect your hand. However, BB probably likes his hand and was raising regardless.

Checking behind on the flop isn't bad here, as it induces the villain to bet the turn. If he has just a lone K or one pair, you can push any broadway/ace and have a lot of fold equity to go along with your draw.

I bet the flop most of the time even when I have a draw, but this flop seems to be a good one to check behind simply because top pair or better will probably raise to protect against the draws.