The Yankees have yet to make any changes to their starting rotation and it seems like they're unlikely to do so any time soon. Brian Cashman has admitted that the rotation is already full with Masahiro Tanaka, Michael Pineda, Nathan Eovaldi, Luis Severino, CC Sabathia, and Ivan Nova on the roster. The full (of questions) rotation and lack of adequate pitching depth in the upper minors led them to upgrade their depth, acquiring Luis Cessa, Chad Green, and Ron Herrera through various trades in the offseason. Still, it would be a good idea for the Yankees to sign a veteran pitcher on a minor league deal.
Last year's Triple-A rotation was comprised of current free agent Kyle Davies, who was never going to be a real major league option, Jaron Long, who was demoted, Luis Severino, and Bryan Mitchell. Maybe you keep Mitchell in the minors and bring Long back up to fill innings. Out of the three pitchers the Yankees have acquired this offseason, Cessa is the only one with any experience at Triple-A. Based on who played where, you could also have Brady Lail, Caleb Smith, Eric Ruth, and Miguel Sulbaran fill out innings in the upper minors.
Unfortunately, with a major league rotation filled with so many question marks, the Yankees need a pitcher they can actually rely on. Brian Cashman probably agrees as well, otherwise he would have just gone with the internal options I listed above. Green and Herrera can probably move to Scranton at some point, but the Yankees should look into signing a veteran pitcher who can pitch decently, provide some injury depth, and then disappear thanks to an opt out clause if the team doesn't need them. They do it every year, and 2016 should be no different.
The Yankees will likely want a guy who is healthy, because they don't need any more risks, as well as someone who is able to give them plenty of work if it ever comes to it. With that criteria in mind, Jeremy Guthrie could be the perfect fit. He's been healthy, reaching the 200-inning plateau twice in the last three years, though struggling badly in 2015. At the age of 37 and off a down year, it's doubtful that anyone would be offering him a major league contract, so he might actually agree to terms with the Yankees.
There is also Aaron Harang, who is a season removed from a 2.5-WAR, 200-inning surprise campaign, though he was much worse in 2015. He's not going to be a staple in the rotation, like he was with the Braves in 2014, but he could offer some solid innings. Then there's Chad Billingsley, who struggled in his first year back from Tommy John surgery, or Joe Blanton, who was actually pretty good as a reliever this year. Mat Latos was also pretty bad last year, though he remained healthy.
Look, it doesn't really matter who it is, as long as they stay healthy and throw innings. It would be nice to look at the Yankees' minor league system and see all these useful starting pitchers ready to make the move up to the majors, but that's not what I see. The Yankees just traded away Rookie Davis, who was promising, but hadn't yet hit the upper minors. They have had several prospects fall apart from injury, and the ones who have reached the cusp, aren't exactly can't-miss guys. Mitchell has been terrible in the majors so far, Lail had a strikeout rate in the 5.0 region, and who knows what kind of pitchers Smith and Sulbaran will become.
In a few years they'll have a better idea of who some of these guys are and recent draft picks like Ian Clarkin, James Kaprielian, Jordan Montgomery and the rest of the college arms will be ready for the majors. Unfortunately, 2016 is here and right now things are looking rather barren. Severino graduated and that's an amazing thing, that's what you want to happen. However, right now, especially if they're not adding a substantial upgrade to the rotation, the Yankees need a veteran to provide some stability. The hope is you never see him in the majors in any kind of significant role, and if Nova is going to pitch out the bullpen until someone gets hurt, you won't. Just like what's happening with the Dodgers right now, you can never have enough pitching. You just have to snatch it up first.