Looking at the map of the United States there is a strong correlation between union density, right to work and presidential voting. Historically the right-to-work states are the Confederate States of America plus the Great Plains and Mountain West (minus Montana, Colorado and New Mexico). These are all states generally speaking that have the lowest union density and vote for the Republican presidential candidate.
There are anomalies. For instance, Nevada is a right-to-work state and is in the top 15 in terms of union density, outstripping traditionally strong union states such as Pennsylvania and Ohio. But this can probably be explained by the importance of Las Vegas to the state's economy and the historic role of organized labor in the development of that city. (Another anomaly is Alaska, which has one of the highest rates of union membership, is not right to work and hasn't gone to a Democrat since LBJ in 1964. Not to be glib but having traveled to Alaska it is easy to conclude that it is a world unto itself. There is a heavy military as well as oil industry footprint.)
All in all it's a reliable formula. You want to drop union density? Pass a right-to-work law and gradually the state will turn Red on the electoral map. Thinking about my own office workers bargaining unit, if people were given the option not to pay dues, at first there might be only one or two who would choose not to; but over time that number would certainly increase. Idaho passed right to work in the 1980s and its unions have steadily eroded. Last year the Idaho legislature amended the law to make it tougher, banning even the use of market recovery money. Market recovery is when unions subsidize a signatory contractor so that it can be competitive in the bidding process with non-union companies.
Reading Monica Davey's story this morning about Michigan going right to work, her description of how the Republican Governor Rick Snyder immediately and on the down-low signed the bills into law and then announced it to the media after the fact, the whole thing -- state troopers facing off against union protesters outside -- has the vibe of a putsch.