How can a state court avoid review of a case by the US Supreme Court?

The only state courts eligible to have cases reviewed by the US Supreme Court are the state supreme courts or court of appeals immediately under the state supreme court (if that court declines review), because a case must exhaust all appellate options before any party can petition the US Supreme Court for a writ of certiorari.

State supreme courts can avoid having their cases reviewed by never accepting a case that involves questions of federal or constitutional law. Barring that, they could pay careful attention to case law and not make a ruling in conflict with an established precedent.

Barring that, they can only hope whoever loses the state supreme court appeal doesn't petition the US Supreme Court for a writ of certiorari or, if they do, hope that the case isn't sufficiently compelling that it attracts four of the Justices' interest.