Aliens All Speak English...
This comes up as a gripe about Star Trek often, and it is annoying. I have looked through Marc Cushman's book "These Are the Voyages - Season One" to find references to the problem in early Trek.
In production order:
Balance of Terror - "The episode is marred by a notable flaw: The Romulans speak English amongst themselves. NBC was not likely to agree to sub-titles, nor was there time or money to devise a language for the aliens." - The reference to this episode comes from Marc Cushman's assessment of the episode. I include it only because it mentions the inherent problems in using an alien language.
Miri - "Roddenberry and Spies were in immediate agreement that Miri’s world should be described as “another Earth.” This allowed for contemporary Earth-like locations and clothing to be used, thereby saving money. It also explained the use of English as the children’s language. And it was hoped this would better help the TV audience to empathize with the children -- contemporary American Earth children." - This is an unfortunate early example of the shallowness of Roddenberry's appreciation of science fiction, "another Earth", being a totally silly notion. But it also points out the practical reasoning that was inescapable in '60s TV.
Arena - "Roddenberry wrote: [Regarding] Kirk’s voiceover during battle -- obvious why you had to go this direction. Good voiceover; well handled. But let’s discuss a couple other possibilities, i.e., a) the Gorn can speak English after some effort...(the other possibilities were ways to dodge the Gorn speaking at all)" AND "Under Gene Coon’s guidance, Star Trek was rapidly becoming faster-paced. Coon felt that Star Trek could remain adult and, at the same time, be a bit more fun. Care, however, was still taken to keep the series believable. To this end, it was agreed among the creative staff that the Gorn would not speak English. Instead, to honor a suggestion made by Roddenberry, a translator device was introduced to allow the two combatants to communicate." - Again we see Roddenbery struggling with the problem... And, interestingly, a solution generated which made sense. The translator was an advanced Metron technology, however, and no Universal Translator had yet been conceived.
Gene Coon on Errand of Mercy - "We are again asked to accept the troublesome tendency of science fiction of this era to present stories on other worlds where everyone, planet natives as well as invaders, speak English, even when not in the presence of those who normally communicate in that language." - So, the worry was growing.
Unfortunately it would never be a solved problem. A Universal Translator device was introduced in Meatamorphosis (season 2) but not referred to again till the animated series. Fans and novelists often assumed that the ship's computer and communicators linked to it handled translation, but this was not in the shows. By The Motion Picture, subtitles were acceptable with alien spoken language, but TNG rarely mentioned the problem, although one of the best episodes . Its spiritual descendant, The Orville, casually wrote the technology in and mentions it fairly often...
One additional note (From Memory Alpha Wiki) Jerry Sohl, the writer of "The Corbomite Maneuver", later explained, "We were originally going to have [each crew member] carry a language translator, which would fit on the wrist like a beeper, and no matter what area of the universe they were in, the thoughts that the people were thinking would automatically be translated into English as they spoke. We got rid of that idea, and assumed that everybody did speak English." (The Star Trek Interview Book, pp. 127-128)