Now, I love Star Trek in most all of its reincarnations, but the Red Shirt trope is so bad that whenever someone we don't know gets involved in anything, we all go "uh-oh. He's toast." You see it a lot in other science fiction and in fantasy, thrillers, action movies, horror, etc. It's also cleverly spoofed in many good parodies.
I have to admit, though, that I prefer the "Red Shirt" problem to the "Everyone's a Red Shirt" phenomena in grim works like Game of Thrones or Shakespeare tragedies, where no one's safe and anyone can die. Readers and viewers grow attached to their favorite characters. They don't want them dying all over the place, even if it's more realistic.
Click on tvtropes.org to see their classic photo of a Red Shirt in its natural state. And as this great website warns in its entry on Mauve Shirts (those Red Shirts who have had good lines or enough personality for us to get slightly attached), whatever you do, if you're a Red Shirt, don't show anyone a picture of your baby. You'll die for sure then.