Let us begin with this simple, indisputable truth: public employees' unions don't get a single red cent from taxpayers. And they aren't a mechanism to “force” working people to support Democrats – that's completely illegal.
Public sector workers are employed by the government, but they are private citizens. Once a private citizen earns a dollar from the sweat of his or her brow, it no longer belongs to his or her employer.
In the case of public workers, it is no longer a “taxpayer dollar”; it is a dollar held privately by an American citizen. Public sector unions are financed through the dues paid by these private citizens, who elected to be part of a union – not a single taxpayer dollar is involved, and no worker is forced to join a union against his or her wishes.
No worker in the United States is required to give one red cent to support a political cause he or she doesn't agree with.
There is no distinction between the role public- and private-sector unions play: both represent their members in negotiations with their employers. At the federal level, both are prohibited from using their members' dues for political purposes.
They donate to political campaigns – to elect lawmakers who will stand up for the interests of working people – but only out of voluntary contributions their members choose to make to their PACs.