Is the US Supreme Court pro-employer? Or pro-employee?

 Neither one, in my opinion. The Justices don't wake up in the morning thinking "How can I help employers?" Or "How can I help employees?"

What they do is interpret Congress's statutes the way Congress wrote them.

 Recent examples: 

 > Murray v. UBS Securities (9-0) [PDF] – A Sarbanes-Oxley Act whistleblower must prove that their protected activity was a contributing factor in the employer’s unfavorable personnel action, but need not prove that the employer acted with “retaliatory intent.”

> Muldrow v. City of St. Louis (9-0) [PDF] – An employee challenging a job transfer under Title VII must show that the transfer brought about some harm with respect to an identifiable term or condition of employment, but that harm need not be "significant" or "serious" or "substantial" or any similar adjective.

> Bissonnette v. LePage Bakeries (9-0) [PDF] – A transportation worker need not work in the transportation industry to be exempt from coverage under Section 1 of the Federal Arbitration Act.

> Smith v. Spizzirri (9-0) [PDF] – Section 3 of the Federal Arbitration Act requires district courts to stay a lawsuit pending arbitration; no discretion to dismiss the case.

> Starbucks v. McKinney (9-0) [PDF] – Courts must evaluate the NLRB's requests for injunctions under Section 10(j) of the National Labor Relations Act using the traditional four-factor test for preliminary injunctions, not some other more lenient standard.

> Harrow v. Department of Defense (9-0) [PDF] – The 60-day deadline for a federal employee to petition the U.S. Court of Appeals for the Federal Circuit to review a final decision of the Merit Systems Protection Board is not jurisdictional.

 Your results may vary. If you have any of these issues, consult a good employment lawyer.
(Not me, I'm an arbitrator.)