There are a number of sources outside of the New Testament which indicate that Jesus had a reputation for being a miracle worker. The non-Christian writer has no real reason to create a Jesus that preformed miracles especially of those miracles were intended to validate his message. While the evidence is meager, it is an indication that Jesus’ reputation as a miracle worker was known outside of the Christian community by people that would have no interest in enhancing Jesus’ reputation.
Josephus attributes miracles to Jesus, as does the Babylonian Talmud. In the second century Origen quotes the heretic Celsus as thinking that Jesus learned his magical arts in Egypt, and when he returned to Galilee he did miracles in order to claim to be God (in Contra Celsus, 1.38, cf 1.160). Celsus is late evidence since he is living at the end of the first century and likely only knows Jesus through the Gospel traditions. A rather indirect piece of evidence is that the name of Jesus is associate with healing spells and exorcisms. The evidence for exorcisms is rather late (AD 330), but rabbis forbid using Jesus name in healings as well.
The so called criterion of authenticity can applied to the miracle stories. For example, all strata of the tradition indicates that Jesus did miracles, including Mark, Q, M/L, and John. This ought to satisfy the criterion of Multiple Attestation since miracles appear in all of the various forms suggested by form criticism. Given the methodology of even the Jesus Seminar, one can confidently conclude that Jesus had the reputation as a miracle worker, that he claimed to do miracles, healings, etc.
The criterion of plausibility argues that an event is more likely historical if it is a plausible event. If this is applied to the miracles, many will dismiss miracles because they do not seem plausible. What is or is not plausible is highly subjective, and very often implausible events actually occur. To me, it is implausible that anyone claiming to be a messiah in the Second Temple Period would not do miracles. While the modern worldview would dismiss miracles as implausible, the Second Temple Period would require them if Jesus was to be taken seriously as the messiah!
The criterion of embarrassment is more helpful. If a deed seems like it might have been an embarrassment to the growing theology of Jesus, and they passed it along anyway, it has a greater claim to validity. The healing of the woman with the flow of blood, for example, has Jesus healing the woman without really consciously thinking about it, the power just “went out of him” and he did not know who it was that touched him.
In addition, Jesus was known to have been a man of prayer, yet there are no stories in which Jesus prays in connection to a healing. If the early church were going to create or enhance the prayers of Jesus (which they very well may have), it is remarkable that they did not create prayers to be added to the miracles of Jesus. This means that Jesus did not heal in the same way Jewish holy men healed, through prayer and ritual.
In short, it is historically plausible that Jesus was known as a miracle worker during his own lifetime, even if the modern thinker dismisses the possibility of miracles.