Several Earth languages are mentioned in TOS, but none are specifically identified as Federation Basic.
"Tomorrow is Yesterday":
KIRK: Welcome aboard the Enterprise.
CHRISTOPHER: You speak English.
KIRK: That's right. You can step off the transporter
MCCOY: Well, either choke me or cut my throat. Make up your mind.
KHAN: English. I thought I dreamed hearing it. Where am I?
KIRK: We're real enough.
COCHRANE: You speak English. Earth people?
COCHRANE: What's the theory behind this device?
KIRK: There are certain universal ideas and concepts common to all intelligent life. This device instantaneously compares the frequency of brainwave patterns, selects those ideas and concepts it recognises, and then provides the necessary grammar.
SPOCK: Then it translates its findings into English.
COCHRANE: You mean it speaks?
"The Doomsday Machine":
SPOCK: Impulse and warp engines operative, transporter and communications under repair. Random chance seems to have operated in our favour.
MCCOY: In plain, non-Vulcan English, we've been lucky.
SPOCK: I believe I said that, Doctor.
"Wolf in the Fold":
COMPUTER: Redjac. Source Earth, nineteenth century. Language, English. Nickname for mass murderer of women. Other Earth synonym, Jack the Ripper.
KIRK: Jack the Ripper?
UHURA: The dog has a, The dog, (something)
CHAPEL: Not Swahili, Uhura. In English. The dog has a ball. See? B, ah, ll. Ball. Now you go ahead.
"Bread and Circuses":
FLAVIUS [OC]: Don't move! Hands in the air!
SPOCK: Complete Earth parallel. The language here is English.
"Bread and Circuses":
FLAVIUS: Are you trying to be funny?
SPOCK: Never. Colloquial twentieth-century English. An amazing parallel.
"Spectre of the Gun":
MELKOT [OC]: Aliens, you have encroached on the space of the Melkot. You will turn back immediately. This is the only warning you will receive.
SPOCK: Vulcan, Captain.
CHEKOV: It was Russian, sir. Every word.
UHURA: No, Captain. It was Swahili.
KIRK: Interesting. Telepathy.
These episodes are enough to show that the normal language aboard the TOS era Enterprise is English.
in "The Man Trap":
UHURA: I see. So naturally, when I'm lonely I think of you.
CREWMAN: Ina cuvanea mwanamke turee.
UHURA: Una kafeeri Hur. You're Swahili?
Some French and German is spoken in "The Squire of Gothos":
KIRK: Not yet. Put it on stun, not to kill, DeSalle.
TRELANE: DeSalle, did you say? Un vrai Francais?
DESALLE: My ancestry is French, yes.
TRELANE: Ah, monsieur. Vive la gloire. Vive Napoleon. You know, I admire your Napoleon very much.
KIRK: This is Mister DeSalle, our navigator. Doctor McCoy, our medical officer. Mister Sulu, our helmsman, and Carl Jaeger, meteorologist.
TRELANE: Welcome, good physicianer and honourable sir. (bows low)
SULU: Is he kidding?
TRELANE: Und Offizier Jaeger, und der deutsche Soldat, nein? (gives a little Prussian salute then marches around) Eins, zwei, drei, vier. Gehen vir mit dem Schiessgewehr.
Since Desalle said that his ancestry was French, it is quite possible that he didn't speak French himself. For reasons of diversity it would be better if Jaeger was German born, but I don't have any evidence whether Jaeger spoke German.
So English is established as the normal working language aboard the TOS era Enterprise, English is established as Kirk's native language, the extraterrestrial language Vulcan is established as Spock's native tongue, Russian is established as Chekov's birth language, and Swahili is established as Uhura's native language.
In the era of TOS, at least three languages, English, Russian, and Swahili, are still people's native languages on Earth. Since only one of them could be Federation Basic, there are at least two other languages in daily use on Earth in the era of TOS.
So the question is whether that situation changed in the century between the TOS era and the era of TNG.