Ultimately, the manager bears the overall responsibility for wins and losses.
That said, I am more and more convinced that the biggest problem with this team is the departure of Ortiz, over which Farrell had no control. The simple fact is that last year the hitting was terrific and this year it's so-so which experience says is simply not going to hack it for a Red Sox team. Ortiz was possibly the best hitter--certainly as measured by OPS--in MLB last year. That's significant, but it now appears that his bat in lineup lifted others' hitting and without his bat HanRam, Betts, Pedroia, JBJ, et al have lower OPS's than last year. Moreland, whose bat replaces Ortiz's in the lineup, is actually having a good year for him, but his OPS is 200 points lower than Ortiz's was. On top of all that, Benintendi, the very promising rookie who skipped AAA last year and seemed to be comfortable hitting (OPS .835) for Boston, has so far disappointed (OPS .711) this year as a regular in LF. We were all disappointed with the 3b situation last year, but this year Farrell has been forced to start Marrero, whose OPS of .562 has to be competitive for the lowest OPS for a regular in MLB, simply because he is a reliable fielding thirdbaseman.
I'm not so much defending Farrell because I believe he bears some responsibility for overall team performance, but am asking the obvious question, what should he have done to fix the hitting which is the real problem with this team?