Well, grammar in Modern English has gone through an interesting evolution. In the Elizabethan Era, grammar was a more subjective affair, growing more and more standardized through the 19th Century in a shrinking world. Conversely people like Mark Twain started writing literature that incorporated regional dialects and deliberately incorrect grammar to develop his characters and narrators....and the uber proper grammar they use. Who talks like that?
If these trends continue 23rd and 24th Century literature will still surely feature artistic license, but written and spoken language may go in a more formally correct direction, or perhaps a more homogenized direction. In my view, language is first a tool for clarity in communication and a signature of cultural heritage first, an example of form following function.
Anyway, one of Harlan Ellison's concepts in "The City on the Edge of Forever" was that language is in a constant state of evolution, so for Kirk, Spock, and McCoy to travel back to the 1930s, they'd surely encounter a language that seemed at least a little foreign to them.
Not to mention the fact that most of the characters we encounter are the elite minds of their respective Academy classes, or at the very least, intelligent enough to operate starships. Surely this would increase the likelihood they'd speak at least their native tongues, or the predominate tongues of their ships properly.
Or maybe the Universal Translator simply edits everyone for clarity's sake?