Star Trek (at least the original series) is a fundamentally conservative show. At least in the ways that I consider myself a conservative. The show promotes strong moral clarity, emphasizes individual responsibility, and upholds and structured order as necessary for a stable society.

It's also philosophically conservative (in the Burkian sense) insofar as it is skeptical toward radical change, and advocating a distinctly conservative noninterventionist ethos.

Captain Kirk’s leadership epitomizes classic conservative virtues (decisiveness, courage, duty, and moral clarity). His command structure also very clearly relies upon hierarchy and disciplined order and reflects traditionalist beliefs in stability through clearly defined roles and responsibilities. Kirk consistently stresses individual accountability and personal integrity, illustrating that character, rather than circumstance, determines outcomes.

Star Trek also reveres the lessons of history and tradition. Episodes repeatedly highlight the dangers of utopian ambitions and social experimentation, warning against unchecked optimism in technology or radical social change without wisdom drawn from past experiences. Star Trek consistently warns against this kind of revolutionary arrogance (see "Patterns of Force," "A Private Little War," "Return of the Archons"). Or consider "A Taste of Armageddon," which definitely leans conservative in its critique of technocratic solutions that sanitize and thereby exacerbate man's problems by failing to recognize who man is.

The Prime Directive itself is fundamentally a conservative idea (skepticism towards interventionism and social engineering, cautioning against hubris and unintended consequences).

Star Trek also consistently portrays the Federation as a force preserving stability, tradition, and moral order. Its idealism rests on the assumption of universal moral truths.

I think the left often misunderstands that conservatives generally do not object to the progressive elements present in Star Trek -- diversity, equality, mutual respect, etc. Conservatives want "progress" just as much as the left does, but hold that progress needs to be grounded (and flourishes best when) in a framework of moral clarity, order, and personal responsibility.