Voter turnout in previously noncompetitive states would increase.
Nonprofit Vote has some figures that back you up.
http://www.washingtonpost.com/blogs/the-fix/wp/2013/03/12/the-states-with-the-highest-and-lowest-turnout-in-2012-in-2-charts/Swing states tend to have high turnout, although there are exceptions and interesting further details (Minnesota has the highest turnout, New Mexico's in the bottom ten, Conservative southern states had low turnout possibly due to dissatisfaction with both tickets.)