Michael Westmoreland-White sees the Star Trek universe as a liberal or progressive vision of the future, featuring things such as racial and gender equality, free universal health care, and an absence of imperalism. I'm neither liberal or progressive, but I think that he's right.
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But I do think that Star Trek is a fairly progressive/liberal science fiction franchise. It’s a basically hopeful vision of the future. It offers up a future earth that has survived war, terrorism, and ecological disasters and forged a global government of representative democracy (we are never told this, but it must be some form of federalist system to avoid tyranny). Hunger and poverty have been overcome. Most diseases have been conquered and high quality universal healthcare is available for all. Education is free and the world is highly literate with most people going beyond secondary education. It’s a clean energy society that is eco-friendly. (In Star Trek IV, the Enterprise crew in their stolen Klingon ship actually go back in time to the 20th C. to keep whales from going extinct–and in the process save the earth of their future.) There is finally global racial harmony. And, despite the micro-mini-skirted uniforms that reflected the fact that the original series was made in the ’60s, we finally have gender equality, too.
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Chekhov shouldn't have been there at all - he wasn't actually recruited until something like the second season. Spock and Uhura - it just cheapens her character to have her doing her senior officer. They took away some of her independent spirit. Of course, you could consider that a liberal influence after the whole Monica Lewinsky thing.