The consensus is pretty strong that the use of the term generation in Matthew 24:34 refers to those alive at the time.
Thayer's Greek Lexicon
STRONGS NT 1074: γενεά
γενεά, γενεάς, ἡ (ΓΑΝΩ, γίνομαι (crf. Curtius, p. 610)); the Sept. often for דּור; in Greek writings from Homer down;
1. a begetting, birth, nativity: Herodotus 3, 33; Xenophon, Cyril 1, 2, 8, etc.; (others make the collective sense the primary significance, see Curtius as above).
2. passively, that which has been begotten, men of the same stock, a family;
a. properly, as early as Homer; equivalent to מִשְׁפָּחַה, Genesis 31:3, etc. σῴζειν Ρ᾽αχαβην καί τήν γενεάν αὐτῆς, Josephus, Antiquities 5, 1, 5. the several ranks in a natural descent, the successive members of a genealogy: Matthew 1:17 (ἑβδόμῃ γενεά οὗτος ἐστιν ἀπό τοῦ πρώτου, Philo, vit. Moys. i. § 2).
b. metaphorically, a race of men very like each other in endowments, pursuits, character; and especially in a bad sense a perverse race: Matthew 17:17; Mark 9:19; Luke 9:41; Luke 16:8; (Acts 2:40).
3. the whole multitude of men living at the same time: Matthew 24:34; Mark 13:30; Luke 1:48 (πᾶσαι αἱ γενεαί); ; Philippians 2:15; used especially of the Jewish race living at one and the same period: Matthew 11:16; Matthew 12:39, 41f, 45; Matthew 16:4; Matthew 23:36; Mark 8:12, 38; Luke 11:29f, 32, 50; Luke 17:25; Acts 13:36; Hebrews 3:10; ἄνθρωποι τῆς γενεάς ταύτης, Luke 7:31; ἄνδρες τῆς γενεάς ταύτης, Luke 11:31; τήν δέ γενεάν αὐτοῦ τίς διηγήσεται, who can describe the wickedness of the present generation, Acts 8:33 (from Isaiah 53:8 the Sept.) (but cf. Meyer, at the passage).
Some commentaries on Matthew 24:34:
Ellicott's Commentary for English Readers
(34) This generation shall not pass . . .—The natural meaning of the words is, beyond question. that which takes “generation” in the ordinary sense (as in Matthew 1:17, Acts 13:36, and elsewhere) for those who are living at any given period. So it was on “this generation” (Matthew 23:36) that the accumulated judgments were to fall.
Barnes' Notes on the Bible
This generation ... - This age; this race of people. A generation is about 30 or 40 years. The destruction of Jerusalem took place about forty years after this was spoken.
Gill's Exposition of the Entire Bible
Verily I say unto you, this generation shall not pass,.... Not the generation of men in general; as if the sense was, that mankind should not cease, until the accomplishment of these things; nor the generation, or people of the Jews, who should continue to be a people, until all were fulfilled; nor the generation of Christians; as if the meaning was, that there should be always a set of Christians, or believers in Christ in the world, until all these events came to pass; but it respects that present age, or generation of men then living in it; and the sense is, that all the men of that age should not die, but some should live.
Geneva Study Bible
Verily I say unto you, This {t} generation shall not pass, till all these things be fulfilled.
(t) This age: the word generation or age is here being used for the men of this age.
Expositor's Greek Testament
Matthew 24:34 Solemn assurance that the predicted will come to pass.—πάντα ταῦτα is most naturally taken to mean the same things as in Matthew 24:33, he main subject of the discourse, the impending destruction of the Jewish state. Jesus was quite certain that they would happen within the then living generation (ἡ γενεὰ αὕτη), not merely through miraculous foresight but through clear insight into the moral forces at work.
Bengel's Gnomen
Matthew 24:34. Γενεὰ, generation) sc. an age of men. This notion, which agrees with the event, corresponds most properly with the question, when? etc., proposed in ver, 3; cf. Matthew 24:15; Matthew 24:20, ch. Matthew 23:36; Luke 23:38. From the date of this prediction to the destruction of Jerusalem was a space of forty years, and from the true year of our Lord’s nativity to that event was a space of about seventy-five years. The Jews, however (as, for example, in Seder Olam), reckon seventy-five years as one generation, and the words, οὐ μὴ παρέλθῃ, “shall not pass away,” intimate that the greater part of that generation, but not the whole of it, should have passed away before all the events indicated should have come to pass. The prediction is true with respect to either the forty or the seventy-five years.[1062] So accurately did the Evangelist describe it many years before the event took place.