One of the things about navy ranks, especially when you look at eras like the age of sail, ship crews were relatively small (compared to the army) and didn't really have the need for officers and warrant officers to show rank all the time.
The crews were more intimately familiar with each other. I served on a ship with about the same number of crew as the starship Enterprise, and I knew just about everyone on board, including their rank, throughout the majority of my time there. You really do get to know your crewmates...even if it's only their name, rank and face.
So, while Uhura may often wear the short-sleeved tunic, the crew know her (and her rank) without needing to see it on her uniform every day.
Your Friendly, Neighborhood, Navy Vet!
The United States Navy: Life, Liberty and the Pursuit of all Who Threaten It!
"I can imagine no more rewarding a career. And any man who may be asked in this century what he did to make his life worthwhile, I think can respond with a good deal of pride and satisfaction: 'I served in the United States Navy.'" -President John F. Kennedy