In Harris v. Quinn, the Supreme Court upheld the right of state governments to collect union dues on behalf of public-sector employees. However, the Court ruled that state governments cannot do so for domestic workers - those who care in-home for children, the elderly, the disabled, etc. - despite the fact that in several states domestic workers are state employees.
The in-home care industry is plagued by low wages and high turnover. This ruling drives a wedge between domestic workers and workers in other industries, by saying that collecting union dues violates domestic workers' First Amendment rights but not the First Amendment rights of other workers. A large proportion of domestic workers are women and persons of color. They will be harmed by this ruling. More details here.
Although I disagree with the Court's decision in Burwell v. Hobby Lobby Stores to allow Hobby Lobby not to pay for contraceptives, and I do think it potentially sets a terrible precedent, it's probably not that bad because employees might be able to get contraceptives anyway.
I'm no labor lawyer, but for the uninitiated, a "right-to-work law" allows employees to work at a job without paying union dues, if they're not a member of the union. In states without right-to-work laws, workers can be required to pay union dues as a condition of employment.
Letting non-union employees work without having to pay union dues: that sounds reasonable enough, right? Let me try to advance a simple argument for why that's not the case:
Your employer sets your terms and conditions of employment - things you must do in order to keep your job. Hopefully your employer is a benevolent overlord, but ultimately they are looking out for the company bottom line, not for your well-being as an employee. A labor union, on the other hand, does have your well-being as a worker at heart. So why should they not also be allowed to set terms and conditions of your employment, since unlike your employer, the union actually exists in order to represent your interests? In the short run this might cost you more, but the more participation unions have, the stronger they are and the more bargaining power they hold. It stands to reason more employees will choose to join the union if they are paying union dues either way. Mandatory payment of union dues as a condition of employment is one solution to the free-rider problem: non-union employees do not contribute to the union, but they still benefit from the union's collective bargaining.
(This system, where an employer can hire anyone without regard to union status, and employees need not join the union, but must pay fees to the union to cover the costs of collective bargaining, is known as an "agency shop." It is distinct from other systems such as the "closed shop." In a closed shop, the employer agrees to hire only union members, and any employee who quits the union must be dismissed. The legal status of these systems varies; in some locales, closed shops may be illegal while agency shops may be allowed. These systems are both examples of what is known as a "union security agreement.")
ETA: I do not mean to imply that Hobby Lobby would actually be paying for the contraceptives in any case; the insurance company pays for them. But if Hobby Lobby files the paperwork to be exempted from the contraceptives mandate, they will not even have to reimburse the insurance company for the contraceptives. The insurer will either eat the cost or will receive a federal government subsidy.