From what I know about Byzantine history, they were losing to Arabs/Muslims because of their internal instability, rather than because of the Arab military competence.
Byzantines (/East Romans) could have stopped them if they had got their act together and focused on defending their borders rather than on internal squabbles over absurd theological differences. They were wasting their energy and resources on internal power struggles in the time they needed to fight the external enemies. It's tragic, really, that the whole Middle East was turned into such a mess because of the Byzantine incompetence and neglect.
Anyway, if the Byzantines had somehow defeated the Arabs and kept them out of their territories, they'd probably turn East, conquer the remnants of Persia and some territories in Central Asia.
After that it would be the same old story: East Roman Empire fighting endless wars with now-Muslim Persia. Islam would have never spread to North Africa or Anatolia, Western Europe would remain a cultural backwater for much longer period of time (no ancient/classical knowledge transmited via Islamic learning), Spain would look entirely different etc.
Muslims would not have controlled most trade routes, so it's a distict possibility that the European Age of Discovery, which was originally driven by the need to find alternative trade routes to India, would have been delayed.