There are two parts to the controversy over Twins making an offer, and potentially signing, Jarrod Washburn. The first is whether the Twins can use another veteran starting pitcher. The second is whether Washburn has value in that role.
Right now the Twins appear to have four starters who are set, Pavano, Baker, Blackburn and Slowey. Barring injuries or a spring training meltdown, those four are going to be in the rotation. Of course it is possible the Twins would trade one of them, but that seems unlikely. That leaves one opening - presumably the one Washburn would fill.
The other folks currently under consideration for that spot are Liriano, Perkins, Duensing, Manship and Swarzak. Liriano and Perkins are the most proven of that list and both struggled mightily at the end of last season. It appears the Twins have all but given up on Perkins and are looking to move him somewhere else. Liriano has never been the same since coming back from injury. He certainly has a tremendous upside, but there is a reason he lost his spot in the rotation at the end of last season. Swarzak and Manship also both struggled last year and have yet to show they belong in the big leagues, much less in a championship team's rotation. Only several failures and/or injuries would give them serious chances. That leaves Duensing who pitched very well for a brief period last summer. A lot of fans are hanging their hat on Duensing and there hopes on Liriano, but adding Washburn would provide a lot more certainty.
Of course the question is certainty of what. Which brings us to the second part of this discussion. If the Twins want/need another veteran is Washburn the guy. He had a good year last year at Seattle that ended badly in Detroit. Its obvious that Safeco is a pitchers park, so you can't expect the same results at Target Field. But Washburn had good success even when compared to other Mariner pitchers. There is not much reason to think he won't be a serviceable starter in the 5th spot in the rotation. As the saying goes, you can't have too much pitching. Adding Washburn gives the Twins the depth to choose the best pitcher available instead of the last guy standing, which is where they ended up toward the end of last season.