Great question. I would honestly recommend starting with the resurrection of Jesus, since that is the ultimate miracle that Jesus based His claims upon (Mt. 12:39-40; Jn. 2:19-21). If Jesus rose from the dead, then it is certainly likely that He could perform miracles like stilling a storm, casting out demons, etc. See http://www.carm.org/did-jesus-rise-dead-outline.
Furthermore, to even take a step back behind the resurrection, you could even start with the existence of God. If God exists, then miracles are possible. In fact, the greatest miracle to ever occur was the creation of the universe from nothing where all space, time, and matter had a beginning at a point in the finite past. If space, time, and matter could come into existence through the causation of a Being, then such a Being could act in miraculous ways: hence a resurrection, stilling a storm, etc.
I think that the most critical issue regarding the miracle claims in the Gospels is not so much about what external corroboration that we have for them, but the worldview through which one views those events. If God exists, then miracles are possible. If Jesus rose from the dead, then why not believe that He also performed miracles?
Also, another way to approach the issue is to look at the Gospels as reliable historical documents who accurately record history and distinguished between fables and truth (2 Pet. 1:16), contain eyewitness testimony (Luke 1:2; cf. 2 Pet. 1:16). Since these Gospels report Jesus performing miracles, we should not dismiss them as unreliable records.
This is just a start to how I would approach the issue.
God bless you!