In a word, no.
It Is High... and Yaros will disagree with me, and both utilize the same key philosophy: There's nothing to lose. Both call the signing a "no brainer," as he'd be coming at a league minimum after the Dodgers bought him out.
But we don't need him. We really don't. There's no reason to jump after fading (faded?) stars just because there's "nothing" to lose. Andruw has no work ethic whatsoever, and his anemic hitting last season was worse than Melky Cabrera's. I'm not a fan of Melky, and I think Brett Gardner is just a placeholder for Austin Jackson (eventually), but both are better stopgaps at this point than Andruw.
He's old. He stinks. He has a bad attitude. So he's cheap. Big deal. We already have too many outfielders, and there's no reason to bring in more.
Yes, we did bring in Sidney Ponson last year despite his troubled ways. But Ponson was necessary. We were desperate for pitching. We aren't desperate for outfield depth, not with Melky, Gardner, Xavier Nady, Nick Swisher, Johnny Damon, and the occasional Hideki Matsui sighting.
Andruw's not as fast as he used to be (thanks to his weight problems), so he's not as good a fielder anymore. He doesn't have the same bat speed, so he can't hit anymore. He's simply an all-around lousy player at this point. There's a reason why the Dodgers were so apt to dump him. They ate a lot of money to cut this guy.
Do the Yankees have anything to lose? No. But do they have anything to gain? Nope.
Is it possible for him to turn it around? Well, anything is possible. But the signs say it won't happen. He didn't just magically stop being good. He stopped trying, he gained weight, and he aged a little. Combine those three, and voila! A wrecked career.
The Yankees have no need to try him out, and they therefore have nothing to gain by signing him. There is a remote chance of a turnaround, but it's unlikely. All that would do is further clog the outfield. If he was a starting pitcher, maybe-- hell, the Yanks have been mentioned alongside Freddy Garcia this offseason. But we don't need an outfielder, even if there's nothing to lose. That's not a good enough reason to sign somebody.
In a word, no.