There is no proof. I do think the balance of the evidence favors the standard model of Jesus' historicity, at least on the broad outlines, than otherwise, although it isn't something to be dogmatic about (there is much room for doubt). It is rather akin in my mind to the question of the existence of the Teacher of Righteousness, the founder of the Qumran sect of the Essenes. There probably was a founder for this well-defined religious group (remarked even by Pliny as residing by the Dead Sea), and the Dead Sea Scrolls indicate both the time when he was active (the 150s BC IIRC) and his teachings. But such a figure is not mentioned in any contemporary sources unlike he is identified with any of a number of other figures in the Hasmonean period. Still most Qumranic scholars accept that such that person probably existed.
I am inclined to think that the Christian movement had its start within the popular movement of John the Baptist (to whom Josephus refers); the book of Acts at least shows that many in the early Jerusalem church (helmed by James the Just) had been members of that movement. (John the Baptist, in turn, can be situated quite well within popular Essenism) John was then executed by Herod Antipas and his movement did not come to a sudden stop but seems to have been absorbed within the later Christian movement. It makes good historical sense to think that after John died, someone else stepped into his shoes, took the movement in a new direction by claiming to be a messiah (and Josephus shows that messianic pretenders were quite common in the period), and then was executed by the local Roman government for sedition (again not unusual). Jesus did not need to develop his own movement from scratch but simply drew on an already-existing one, and since John was popularly believed to be Elijah, there was already a biblical model for a successor (with Jesus as an Elisha figure.....hence the miracle stories in the gospels which largely seem to draw on OT stories about Elisha). Then after Jesus' death the movement continued under the leadership of James and the apostles John and Peter (the pillars mentioned by Paul), which sought to work out the meaning and implications (both eschatological and soteriological) of Jesus' death, shifting the focus from halakhic matters of how to achieve true righteousness (the primary concern for John the Baptist, the Jesus of the synoptic gospels, and to a great extent what we know about James) to what Jesus achieved as messiah through his death and resurrection (which is what became the primary concern for Paul). And then Paul transformed the movement further, spreading his own form of this religious tradition into the West to a largely non-Jewish population, where it drifted further from its Palestinian Jewish roots (with much friction resulting between Pauline Christianity and more traditional Torah-observant Christians). Paul was quite clear that he was somewhat of a Johnny-come-lately and that there was already in existence a Jesus movement before him.
So my view of things comes more from the perspective of accounting for the social history of religious movements, which leave a larger historical footprint than any single individual. I think the question posed by the mythicists is whether it is necessary to posit Jesus' existence in order to explain the origin and spread of Christianity. There is no denying the mythical dimension of much of what early Christians believed about Jesus, such as what we find in Paul and in proto-gnosticism, and there is no doubt that the gospel writers invented a lot of the narrative by drawing creatively on OT material. But I do not think it was necessarily "turtles all the way down" because real people did claim to be messiahs (such as Simon bar Kochba) and it was the office of the messiah that drew on a host of mythical traditions about messianic identity and function. One can readily recognize how the ideas surrounding Jesus Christ in the NT incorporates many layers of messianic tropes already present in pre-Christian messianic Judaism. For example, Jesus as the eschatological heavenly "Son of Man" directly borrows the Enochic conception of the "Son of Man" and heavenly Wisdom found in the Essene Book of Parables from the early first century AD. Qumran messianic texts have expectations about various future end-times priestly and royal figures (CD, 4Q246, 4Q252, 4Q521, 4Q534, 4Q540-541, 11QMelch) which have strong parallels to ideas found in the NT regarding Jesus. These and many other notions drawn from the OT (such as the messiah as the Suffering Servant figure from Deutero-Isaiah, as a Moses figure, etc.) were applied to Jesus by those who believed that he was the messiah. But what is striking is that in all pre-Christian eschatological texts about the messiah, the messiah is an imagined future deliverer, whereas the earliest Christian texts look upon Jesus as a messiah whose activity lay in the past (and who continues to be active in the present). The only Jewish parallel that readily comes to mind is John the Baptist, who occupied the role of Elijah redivivus. There was an established mythic eschatological expectation that Elijah, a heavenly figure, would return at the end of days, and those expectations looked towards the future. The followers of John the Baptist however drew on Elijah mythology as pertaining to the past: "Elijah has already come, and they did not recognize him, but have done to him everything they wished" (Matthew 17:12, Mark 9:13; cf. Luke 9:8, "others said Elijah had appeared" regarding Jesus' succession to John the Baptist). This seems to be quite parallel to what happened with Jesus. It wasn't simply a matter of Jews hoping for the return of Elijah inventing the character of John the Baptist as a mythic Elijah figure to fulfill those expectations; John was a historical person who led a popular Jewish movement critical of the Herodian administration, as Josephus makes clear, but the religious movement came to identify him as Elijah whose return at the end times had been prophesied. In the case of Simon bar Kochba, a clearly historical figure, we can see a similar pattern. He was identified with the "star", a heavenly symbol, from the Balaam oracles in Numbers. The description that Jerome gives of him smoking lighted blades of straw while he talked suggests that Simon tried to identify himself with the messianic "Man From the Sea" figure from 4 Ezra who spoke with a fire in his mouth and sent forth "from his lips a fiery breath" (13:10-11). So too it is conceivable that another messianic claimant, Jesus, could come to be viewed as identified with the Enochic Son of Man, as heavenly Wisdom, as the Logos, as the branch of David, as Melchizedek, as the morning star, as the Moses-like Taheb, etc. Christians seemed to apply every single messianic motif in existence to their messiah. So I do not find that all this mythologizing and theologizing necessarily indicates that Jesus originated as a purely mythological character, and there are indications that a non-mythological human being lies under the surface. Paul loves to talk about Christ in abstract, cosmic terms when he is theologizing about the religious significance and meaning of Jesus, but when he writes more plainly about his history in the Christian movement in his apologia in Galatians, Jesus appears more down-to-earth as the brother of the leader of the Jerusalem church, James (1:19; cf. 1 Corinthians 9:5), and even when he was theologizing he still clearly conceived of Jesus as a Jew who was "born under the Law" (Galatians 4:4) and "descended from David under the flesh" (Romans 1:3-4), and who was killed just as the prophets of old were killed (1 Thessalonians 2:14-16). This does not prove that Jesus was a historical figure; the evidence overall is much, much too slight to serve as proof of anything solid, but it does show that Paul did think of Jesus as a flesh-and-blood person and thus it is incorrect to characterize Paul as conceptualizing Jesus purely as a timeless heavenly figure. I think what can be glimpsed in Paul is the process of mythologizing a messianic claimant as all the things a proper messiah should be; Jesus was to him at the same time the person who was born under the Law and who died and was buried, as well as the heavenly Son of God who pre-existed his earthly life and who rules over all creation by his Father's side. It was the belief in Jesus' resurrection (revealed to Paul, Peter, and others through vision) that allowed Paul and other early Christians to connect the messanic claimant Jesus with the imagined spiritual being in heaven who embodies everything that Second-Temple Judaism expected the heavenly Son of Man/Logos/Wisdom to be.
Anyway, that's just my own hypothesis and opinion....I am pretty agnostic about the matter and whenever I write about the origins of Christianity I always try to focus it more on the community (for which there is a lot more evidence) than an individual, Jesus, who is buried under such a heavy weight of interpretative traditions that I never felt certain that there was someone there underneath it all. Many scholars have sought to reconstruct the "historical Jesus" and their efforts show just how arbitrary and subjective such an enterprise is. But I think there is a good case to be made that there was a historical figure at the font of the Christian movement (just as the Teacher of Righteouness was the founder of the Qumran sect, just as Simon the Galilean was the founder of the Zealot movement, just as Muhammad was the founder of Islam, just as Martin Luther was the founder of the Protestant movement, just as Joseph Smith was the founder of the Mormon movement, etc.), and if religious tradition unanimously names that figure as Jesus, I think it is more likely to surmise that there was a messianic claimant bearing that name who was later mythologized (which is a common pattern in religious movements) than a purely mythological figure who was historicized. But I do not consider this a matter that could be definitively proven, and like many other aspects of history, it is something that can be debated and doubted. The proof just isn't there....at least not with the evidence we currently have.
