It's a question that's been going around a lot this week. With all the conflicting reports on the Yankees' interest in Drew and his interest in the Yankees, it's hard to know where the truth lies either way. The need is obviously there from the Yankees' standpoint. Their infield is currently the biggest weakness on the roster, and Drew would help fix some of those cracks.
The short answer is yes, they should. Drew would be at worst as good a starting third base option as Kelly Johnson, and he would allow the Yanks to move Johnson over to second base, his better position. He could be injury insurance for Derek Jeter at short this season and slide in and take over the job for a year or 2 after Jeter retires. He's not too old, his left-handed swing would play well in YS3, and it wouldn't take crazy money or years to get him.
The smarter answer is no, they shouldn't. If you read between the lines of everything that's been reported this week, as Mike Axisa did at RAB yesterday, you'll see that a lot of it is nothing more than teams and agents trying to gain leverage through the media. The Red Sox are the team with the greatest need for Drew, but they haven't been willing to cough up the dough so Scott Boras is probably leaking "Yankee interest" to the media to attempt to put pressure on Boston. If the Yankees do get drawn into some real talk for Drew, they'll get put in a bidding war with the Red Sox and could end up paying more than they want to for a player who's good but not THAT good.
And the Yankees could very well be feeding false information to the media themselves to keep Drew's cost down. The best thing for them to do right now is stay quiet publicly and deny, deny, deny while trying to work something out with Drew behind the scenes if they are truly interested. If he's serious, see what he's looking for or if he and Boras even want to talk. There's really nothing to lose.