River decision against a fish

[BorisTheSpider]

Hero (CO) ($26.97)
Button ($25)
SB ($11.33)
BB ($25.54)
UTG ($25)
MP ($43.83)

Preflop: Hero is CO with 3d, Ad
2 folds, Hero bets $1, 2 folds, BB calls $0.75

Flop: ($2.10) 3h, Qc, 3c (2 players)
BB checks, Hero bets $1, BB calls $1

Turn: ($4.10) 7h (2 players)
BB checks, Hero bets $2.74, BB calls $2.74

River: ($9.58) Jh (2 players)
BB checks.......


BB is a huge fish, limping all sorts of crap and calling down with any piece. Without the flush card on the river, I'd always bet on the end, expecting to be called by Queen-rag or a 7. When the flush card does come though, do you bet-fold this river for value, or check behind?