Whether they like it or not all depends on their interpretation. The conservatives who didn't like it saw it as an allegory for the communist witch hunts in Hollywood going on at that time. In such a scenario, Gary Cooper was a brave, innocent left-wing person who was being persecuted, and finding people cowardly refusing to help him despite his pleas. They also saw it as violating the conventions of the western, featuring good guys and bad guys. The movie makes everyone in the town look bad, because they're all cowardly. But the conservatives also didn't like Cooper's character seeking help and needing a woman to save him--it made him seem weak, and it also violated the ideal of the independent western man who never seeks help from anyone.

On the other hand, the conservatives who like it appreciate it for the way that Gary Cooper fights injustice himself, even though everyone else turns their back on him. They like his independent spirit to fight instead of run. They see him as being strong and brave.

Liberals who like it see the allegory of the people tormented by the communist witch hunts. Their struggle is glorified through this man, made into a much simpler and obvious fight between good and evil, between bravery and cowardice.

It doesn't look like there was any significant liberal dislike of it. However, communists in the Soviet Union, who aren't part of the liberal American tradition, didn't like it because at the time, the height of Stalinism, collectivism was praised. In their ideal heroic scenario, everyone bands together for a common goal. By emphasizing in this film that one man alone, superior, braver, stronger, better than all the rest, is an ideal worth striving for, the individual was too much praised. There's kind of a common argument that you see recurring in art and literature over time. It's the conflict between the two views of an individual person as good, but people as a collective being evil. Or people together being good, but individuals acting in selfish self-interest.

So which is the correct interpretation? There is none. The sign of a good story is that it is rich and complex enough to support multiple interpretations. I think John Wayne was in the extreme in not liking the movie, though. There was certainly a lot of material in there for conservatives to admire. But doing westerns was John Wayne's whole life. He had a certain way that he felt heroes should be, and that movie plots should be. And he couldn't handle the different way this movie was constructed. I would say in the long run, there are more conservatives who like the movie and rather few who dislike it.It has Gary Cooper's conservative imprimatur on the hero, and the spirit of individuality that is so important to the film, and the spirit of doing right no matter the opposition, is really valued by conservatives--and liberals too, for that matter.

Follow-up question: why did John Wayne collect Gary Cooper's Oscar for him if he so vehemently hated the film? Did he do it through gritted teeth for a friend, did his hatred of the film come later, or some other reason?