gov extracts taxes with threats of prison -> civil "servants" paid with tax money -> unions tax civil "servant" salaries via automatic payroll deduction -> unions use money to "lobby" (bribe) politicians with campaign donations -> politicians set government policy for taxation and civil service
There ARE some rights being violated here, but it is not the civil "servant"'s rights
However, I think the answer is through workplace safety laws (OSHA), fair labor standards act provisions for minimum wage and overtime, ERISA (which sets minimum standards for retirement and health plans), Civil Rights Act provisions, ACA, the Federal Employment Compensation Act, and others.
Unions originated in an era where you could find children working 14 hours in a sweatshop for a starving wage, with locked doors that prevented escape in case of fire, around dangerous machinery that would take off limbs in a blink with victims never compensated and unable to work again.
The case for unions in that era is different than the case for unions today. Worker concerns today, serious as they may be, are categorically different from the concerns of that time. The case for unions today also has to grapple with concerns about rent seeking, keeping dangerous/unfit employees on the job, and reducing job availability or promotion for capable newcomers.
But note that the alternative to unifying workers trade-by-trade or workplace-by-workplace is to just unify enough of the electorate to fix obviously broken labor practices through legislation. It sounds much harder, but we've done it repeatedly, and it doesn't have the same down-sides.
"All Government employees should realize that the process of collective bargaining, as usually understood, cannot be transplanted into the public service. It has its distinct and insurmountable limitations when applied to public personnel management. The very nature and purposes of Government make it impossible for administrative officials to represent fully or to bind the employer in mutual discussions with Government employee organizations. The employer is the whole people, who speak by means of laws enacted by their representatives in Congress. Accordingly, administrative officials and employees alike are governed and guided, and in many instances restricted, by laws which establish policies, procedures, or rules in personnel matters."
That is to say, the place to protect public servants is through ballot, not collective bargaining.
EDIT: Source http://www.presidency.ucsb.edu/ws/?pid=15445