The enforcement of a state court's order would fall to the Office of [insert county of jurisdiction] Sheriff. Sheriff's exist in 48 states in the union, (The office does not exist in Alaska, which has no counties and Connecticut, which has no devolved county government. In addition The District of Columbia and the five inhabited U.S. Territories lack sheriffs due to no County level divisions.). In the U.S., Sheriff denotes an elected (usually, Hawaii and Rhode Island appoint Sheriffs rather than elect tem) head of a police agency that has at least civil law enforcement authority within a county. This means that they are the police agency that will typically enforce warrants and provide bailiffs to court houses in the county, as well as effect arrests for contempt of court charges.
In the U.S., Judges have the authority to charge criminal "Contempt of Court" against anyone refusing to comply with a judicial order. Contempt charges are classified in two sets of categories: Direct vs. Indirect (Did your actions with contempt occur in front of the judge aka in the court room OR out of judicial view AKA out of court room) And Criminal vs. Civil. In the case of Criminal contempt, the charge must be proven beyond a reasonable doubt. But in civil contempt, the charge need not meet this high standard of proof to find guilt and it falls to the party benefitting from the contempt to enforce.
In an case before a state supreme court, typically the cases are all civil (since all crimes, the "plaintiff" is the state, not the victim of the crime), which normally would prevent contempt of court charges from becoming criminal in nature. When the contempt is in violation of a court order, the finding of guilt in contempt allows the Sheriff to enforce any sanctions until the contemptuous behavior ceases. This could be a fine (usually levied per day while the behavior continues) or even jail time (even for civil contempt) until such time as the behavior stops. So in the event of a State Supreme Court being defied by the government, the Judiciary may direct the relevant Sheriff's office to enforce contempt charges against the state officials violating the rules (The U.S. doesn't have any parlimentary governments I.E. where the Speaker of the Lower House has executive authority, so a legislature changing the law so that the judge's decision is moot is perfectly valid. Execution of the law falls to the governor of the state, so the officials in that office will likely be the subject of contempt charges).
A similar system exists at the Federal Level with the U.S. Marshal Service (USMS) which serves the same functions as a Sheriff for the Federal Courts (U.S. Marshals are appointed by POTUS rather than elected, however). If a state was to violate SCOTUS, the Marshals would function in the same capacity of enforcement. They are the civil enforcement division of the Department of Justice (criminal enforcement is the FBI). Like Sheriff's Marshals have deputies to assist the office holder. Deputy (Sheriff or Marshal) denotes a regular employee who is ultimately overseen by an elected or appointed sheriff/marshal (a regular deputy will still have his job no matter the outcome of the next election). In addition, the Sheriff/Marshal office holder can uniquely deputize non-regular deputies to temporarily increase their forces in times of personnel needs. Typically, this can mean an ordinary civilian can temporarily be named a deputy, but also may allow the sheriff or marshal to draft non-deputy local police personnel to assist in their duties (During the 2020 Anthony Brown protests in Portland, Oregon, the Portland Police officers were deputized by the U.S. Marshals Service so that they could assist in arresting protesters who were attempting to burn the Federal Court House after local officials refused to arrest such individuals. All such arrests were thus handled in U.S. Federal courts rather than State Courts.).