When healthy, Francis is a fine pitcher - he limits walks, keeps the ball on the ground, and garners a respectable amount of strikeouts. His 5.00 ERA is a bit unsightly, but his 3.88 FIP and 3.94 xFIP suggest that a fair bit of bad luck was at play, particularly when taken in kind with his solid 2.91 K/BB and 47.0% ground ball rate. The issue here, though, is that Francis hasn't been healthy over the past three seasons, with a problematic shoulder being the primary concern. While Francis did pitch without much discomfort is 2010, Ben Sheets and Brandon Webb are a testament to the potential ramifications of shoulder problems.
Even so, I do think that Francis would be a fine addition to the Yankees. He would likely be expected to be the fourth or fifth starter, and the team would be able to skip his turn every so often as a means to save a bit of wear and tear. It may not be saying much, but I do think that Francis is the best remaining option on the free agent market.