Are there any non-human Starfleet ships?

Question

Does Starfleet or the Federation have any ships where the majority of crew members are non-human?

Or are all Starfleet ships in majority crewed by humans?

Answer 1

Most ships which were shown on screen had the familiar set of most humans, a Vulcan/android/Borg exploring its humanity and a few more or less exotic humanoid aliens. This is understandable as most actors are humans, no wait - all of them are humans (probably?) so this really cut down the filming costs and made it possible for viewers to identify with the characters in ways which would be impossible to do with seven tentacle aliens (again just probably, who am I to judge).

However, you are correct. There are ships with majority of non-human crew:

The USS T'Kumbra was a Federation Nebula-class starship that was in service with Starfleet in the late 24th century. In the 2370s, the T'Kumbra was under the command of Captain Solok. The T'Kumbra had an all-Vulcan crew, similar to the USS Intrepid and USS Hera, which also had predominantly Vulcan crews.

Source: Memory Alpha

You can think of it as if there are various ships with their crews in Star Trek universe but we are just being shown these with most humans and a few aliens.

Also, there is a very plausible explanation (although I am not aware if it was used anywhere in the series). Different species have different biological traits. It would be convenient if not necessary to have the species with similar biology on the same ship to make the life support systems simpler. Although humans can live at 40 degrees Celsius they don't really enjoy it, so if that is a temperature which members of another species consider very cold, don't expect to see them on the same ship with humans. That explains well why would most ships we see have humans and human-like crew members.

Now, why would Vulcans have so many ships of their own when they are so similar to humans? Maybe because they are so annoying? Maybe because they think humans smell bad?

Answer 2

Another answer that tends to be hinted at from time to time is that of the member races of the Federation, that humans are the most prone to loving the adventure, excitement and thrill that comes from exploration of the unknown. As such, they are the most prevalent of those who enlist in StarFleet.

While there are a number of Vulcans in StarFleet, they tend to prefer serving in ships with all Vulcan crews, being around those of like minds, like tastes and like smells. There are also a couple of ships that have all crews of another race. One example from the novels is that there is a science vessel that was designed and built for the Horta, which was dispatched by Starfleet to examine the Dyson sphere where the Enterprise D found Scotty.