The men of the great assembly
According to Berachot 33a, the Men of the Great Assembly (אנשי כנסת הגדולה) instituted blessings, prayers, kiddush, and havdala. (These translations are copied from the linked page at sefaria.org.)
אנשי כנסת הגדולה תקנו להם לישראל ברכות ותפלות קדושות והבדלות
The members of the Great Assembly established for Israel blessings and prayers, sanctifications and havdalot.
The Rambam, in the laws of prayer (1:4), probably based on this source, describes how Ezra and his court authored the blessings because of the inability of the people to express themselves properly in Hebrew.
כיון שגלו ישראל בימי נבוכדנצר הרשע נתערבו בפרס ויון ושאר האומות ונולדו להם בנים בארצות הגוים ואותן הבנים נתבלבלו שפתם והיתה שפת כל אחד ואחד מעורבת מלשונות הרבה וכיון שהיה מדבר אינו יכול לדבר כל צורכו בלשון אחת אלא בשיבוש שנאמר ובניהם חצי מדבר אשדודית וגו' ואינם מכירים לדבר יהודית וכלשון עם ועם ומפני זה כשהיה אחד מהן מתפלל תקצר לשונו לשאול חפציו או להגיד שבח הקדוש ברוך הוא בלשון הקדש עד שיערבו עמה לשונות אחרות. וכיון שראה עזרא ובית דינו כך עמדו ותקנו להם שמנה עשרה ברכות על הסדר
When the people of Israel went into exile in the days of the wicked Nebucednezzar, they mingled with the Persians, Greeks and other nations. In those foreign countries, children were born to them, whose language was confused. Everyone's speech was a mixture of many tongues. No one was able, when he spoke, to express his thoughts adequately in any one language, otherwise than incoherently, as it is said, "And their children spoke half in the speech of Ashdod and they could not speak in the Jews' language, but according to the language of each people" (Nehemiah 13:24). Consequently, when anyone of them prayed in Hebrew, he was unable adequately to express his needs or recount the praises of God, without mixing Hebrew with other languages. When Ezra and his Council realized this condition, they ordained the Eighteen Benedictions in their present order.
Note that Rambam doesn't connect this to the destruction of the Temple; according to him, everyone prayed in his own way (1:3).
This is the source for the prayers having been formulated by the Men of the Great Assembly. However, this tradition (unlike Rambam) says neither that they established the wording, nor that they established eighteen blessings.
A barayta in Berachot 28b tells us that Shim'on Happakuli arranged the eighteen blessings before Rabban Gamli'el in Yavne. (Shmu'el Hakkatan arranged the nineteenth blessing, against the heretics, as told later in the same barayta.)
שמעון הפקולי הסדיר י"ח ברכות לפני רבן גמליאל על הסדר ביבנה
Shim'on Happakuli arranged eighteen blessings in order before Rabban Gamli'el in Yavne (my translation)
The Koren translation at sefaria.org actually tries to harmonize this with the opinion of the Rambam earlier by saying:
Shimon HaPakuli arranged the eighteen blessings, already extant during the period of the Great Assembly, before Rabban Gamliel, the Nasi of the Sanhedrin, in order in Yavne.
Ezra Fleischer, in this article (Hebrew) advances the view that the wording of the prayers was instituted by Rabban Gamli'el at Yavne. Rabban Gamli'el was also the one who scolded Rabbi Yehoshua on "that day" (בו ביום) because the latter said that the evening prayer was optional (Berachot 28a). The mishna in Berachot (4:3,4) shows the differing opinions of his contemporaries: Rabbi Yehoshua thought people should pray a "shortened eighteen blessings" (מעין י"ח), and Rabbi Eli'ezer believed that anyone whose prayer was fixed (קבע) was acting improperly (note, though, that the Talmud interprets Rabbi Eli'ezer differently).
Also, it's obvious that many of the blessings (such as רצה and ולירושלים) must have been instituted after the destruction of the Second Temple. I'm not sure exactly how Rambam would explain these prayers.
Rav and Shmu'el instituted the prayer ותודיענו which is recited in the holiday prayer after Shabbat (Berachot 33b). This is the only explicit statement I know of that refers to adding a prayer in the shmone esre after Rabban Gamli'el.