Flopping middle pair out of the blinds
 

Relatively new to 6 max games and flopping middle pair out of the blinds is my #1 head scratcher right now. A typical scenario I'm encountering at 1/2 and 2/4 tabes:

1 pf limper and two blinds to the flop, pf limper is slightly loose passive calling almost every flop bet typically (35/5)
I;ve got 7-8 in the BB
Rainbow flop of 2-8-J
SB checks to me
Hero????

I find myself betting this flop a lot, I think there's a pretty good chance that I've got the best hand. After the pf limper calls (sb usually folds) I usually have no idea how to play the next 2 streets.

Continue to bet out even when overcards come up? This has been my default play with the idea to fold to a turn raise. Inevitably the ep limper will call my turn bet again. The river comes and it seems regardless of what the card is I end up checking and calling the bet from the ep limper or seeing it checked through.

I know this isn't the optimal gameplan and was wondering what your thought process is for these types of hands. What types of opponents are you betting into and who are you check calling.

Against an aggressive opponent who's 'autobetting' a lot the flop I've been checkraising the flop and leading the turn. Any merit to this play?