In the ANES data set, 58 percent of union members or those living with a union member voted for Obama. If every union member or member of a union household voted as if they were not one and every other characteristic was kept constant, 51.1 percent of them still would have voted Democratic.While Obama still would have won, Enten points out that the 2.8 point difference union members made is "larger than the margin in two of the past four elections, and it’s about the same as it was in 2008." That's important context when you consider a Supreme Court case like Harris v. Quinn, which perhaps hinted at future damage this Court would consider inflicting on unions. And of course, that national percentage of the vote made up by union households isn't distributed evenly across the states—it's a lot smaller in Louisiana, for instance, and bigger in Michigan and Ohio. So as far as the electoral vote in presidential races, or Senate races, unions can help tip the scales. (Additionally, unions have worked to expand their electoral reach by organizing non-union workers in their communities through programs like Working America.)
Obama would have lost 1.4 percentage points off his vote share in 2012 without unions. Instead of his margin of victory over Romney being 3.9 percentage points, it would have been 1.1 points.
And, of course, if you're looking for another reason—beyond simple hostility to workers—for the Supreme Court justices that installed George W. Bush in the White House, or were subsequently appointed by Bush, to be making decisions that harm unions, there it is.