There are many people who would have been involved in the switch, and unless the governor either denied this as a concern (before and/or after review) or intentionally cut corners and understaffed (by his own decision) he can not be the only party responsible.
This may have been a deadly oversight on the people directly involved, or gross negligence. He is however the overall leader and 'the buck stops here' so he bears responsibility of allowing this to happen, or placing the responsibility (directly or indirectly) on someone who allowed this to happen.
Flint has had water issues since 2007, and since switching the water source in 2014, nothing has gone right. Improperly treated water leached led into the city supply and damaged the pipes. Snyder apparently knew or should have known of the problem for a while. Governer Snyder says he has a plan for dealing with the crisis, but many of his plan items are underfunded, and don't address the main issues. So to an extent, he's responsible for some of the problem.
The Flint water crisis is Governor Snyder's fault because nothing has been done to change anything. That is the governor's state and it is the governor's (and entire US government's) responsibility to make sure the people of the state are being taken care of and receiving their basic human rights.
Although Governor Snyder, as the leader of the state government, bears heavy responsibility for what happens in Michigan, he cannot be made to bear the brunt of the Flint water crisis. The city of Flint officials dropped the ball somewhere along the way by not correcting the problem, or demanding state and federal assistance in doing so.
While the governor certainly bears blame, and should be held accountable, it is a reach to blame the entire crisis on one person or position. Governor Snyder can't possibly be held responsible for decades of neglect and mismanagement. He just happened to be in office at the moment when the issue bubbled to the surface, as it were.