I just didn't see the Red Sox lacking fight. I saw a Rays team that was dominating them -- taking advantage of every mistake (and, to me, aggressive mistakes aren't the same as laying down) and giving the Red Sox no opportunities to get things done.
I don't think it's a lack of effort by the Sox that allows the Rays defenders to get to balls and make plays. A lack of effort didn't cause Sonnanstine to handcuff the Sox for 7 innings. A lack of effort by the Sox wasn't the cause of the Rays' hit parade in the 6th inning of that game. And a lack of effort by the Sox didn't make Pena, Longoria, and Aybar hit home runs to put the Sox down early.
Booing, to me, suggests that the person booing thought the Sox plight in Games 3 and 4 could have somehow been changed by playing harder. The Rays dominated those games and that would have been true no matter how hard the Sox played.