I've long been in the trade Josh Beckett camp. I think he'd also welcome a trade because I think he's as sick of the Boston media as the Boston fans are sick of him being a miserable prick (the guy has not only used up all the goodwill from the 2007 championship - he's now running a deficit). Get back a top prospect or two from the Rangers or the Nationals or some team in contention that Beckett gives his OK to and everyone gets to move on with their lives.
Beckett has been tested in the post-season and has produced like no other active pitcher except maybe Andy Pettitte. His numbers aren't great but keep in mind he hasn't given up a HR in his last 10 starts and for his career he's got a 3.48 ERA in 5 starts at Rangers Ballpark in Arlington (where I think he'd be easiest to trade). Beckett has two more years left on his contract at $15 million per year - so he's not even a rental either.
The Red Sox would get a couple of prospects and would allow Franklin Morales to go back to starting where he's been more productive than Beckett. The team would also then have $25 million of payroll flexibility (combining Beckett's and Dice-K's salaries coming off the books) to make the 2013 team more competitive. Plus this year's team would immediately become more likable.
Alex Speier has an excellent breakdown of Beckett's numbers. Too bad he doesn't come right out and say that the Red Sox should trade him now while they have the chance.