I’m sure someone else can give you more of an idea of day-to-day life of Jews under Byzantine rule, but since you're still waiting on an answer, I can give you a little bit of background info on attitudes and laws relating to Jews in the Eastern Empire/Byzantine Empire.

Early Church fathers debated the issue of ‘what to do about the Jews’. For practicality's sake, the 'failure' of Judaism was used as an assertion of the triumph of Christianity. Very early on, the story of the Jews is used to define Christian identity. Paul’s Letter to the Galatians (Chapter 4) tells the story of Abraham and his sons, one born to a mistress and one born to a wife, as heirs to what Paul regards as two covenants – one ‘earthly’ and one ‘heavenly’. Paul sees the ‘earthly’ covenant (bestowed on the son born to the mistress) as the Law of Moses, and the ‘heavenly’ covenant (bestowed on the son born to Sarah) as the one that the Christians are bound by.

Two slightly different views emerged regarding the Jews – one, that they were not deliberate unbelievers and heretics, and two, that they were. Augustine of Hippo, father of the Latin tradition, defined the role of Jews through his Doctrine of Jewish Witness – he basically said that Christianity needed the Jews to stick around in order to show the background of Christianity (and because they were such good librarians!!).

A year before Augustine’s conversion to Christianity in 387, John Chrysostom was ordained as a priest in Antioch. Chrysostom heavily influenced the Eastern Christian tradition and attitude towards Jews, and is kind of the father of Christian anti-Semitism. His homilies denounced the Jews, and he regarded them as heretics (wilful misbelievers) rather than infidels, who were deliberately insulting and denying their biblical heritage, rather than just interpreting it wrongly Living Letters of the Law: Ideas of the Jew in Medieval Christianity by Jeremy Cohen

John Chrysostom wanted to make it clear that Judaism and Christianity were mutually exclusive:

“Where Christ-killers gather, the cross is ridiculed, God blasphemed, the Father unacknowledged, the Son insulted, the grace of the Spirit rejected... In a word, if you admire the Jewish way of life, what do you have in common with us? If the Jewish rites are holy and venerable, our way of life must be false. But if our way is true, as indeed it is, theirs is fraudulent.”

So, to simplify early Church attitudes, the Latin tradition, influenced more by Augustine, originally saw the Jews as a misguided lot who had missed the Messiah train (this view obviously changes due to a number of other factors, including the creation of the Talmud, and the image of Jews as Christ-killers becomes more prolific in the West). The Byzantine tradition, influenced more by Chrysostom, regarded the Jews as deliberate wrongdoers.

As in the Roman Empire, there were functional reasons for anti-Semitism in the Byzantine Empire as well as essential/theological reasons. Jews had a tendency to revolt when their religious rights were violated or they were being oppressed. They were forbidden from Jerusalem for the entirety of Byzantine rule (fourth to seventh centuries), and this was obviously a source of constant tension between Jews and Christians in the area. Under Heraclius, who became Byzantine emperor in 610, the Jews of Galilee joined forces with Persia to take Jerusalem and Galilee. Antiochus Strategos’ account of the Sack of Jerusalem in 614 gives us some insight into Christian feelings at the time:

” Thereupon the vile Jews, enemies of the truth and haters of Christ, when they perceived that the Christians were given over into the hands of the enemy, rejoiced exceedingly, because they detested the Christians; and they conceived an evil plan in keeping with their vileness about the people.”

A couple of decades later, the Byzantines reclaimed Jerusalem, and Heraclius dealt with the Jews through massacres and forced conversions. The problem of Jerusalem was ‘solved’ when a few years later the city came under Muslim rule, and Jews were allowed back in.

Apart from Heraclius’ actions in the seventh century, Jews were allowed to live and work within most parts of the Byzantine Empire without too much persecution until the Crusades. There were still periods of harassment (a decade or two per century) when Byzantine emperors tended to enforce harsher restrictions on Jews, particularly during times of religious upheaval. The rest of the time, they occupied a sort of middle ground between Christians and pagans, even though permanent restrictions on many aspects of Jewish life were maintained.

If you want to read up on the sorts of laws that applied to Jews in the early Middle Ages in both East and West, this page from the Fordham Medieval Sourcebook is pretty useful. You can see as well that, at certain times, Jews experienced greater persecution in the West than they did in the East – for example, circumcision was completely outlawed in the Visigothic kingdom under Recceswinth in the mid seventh century, yet circumcision was only prohibited in the Byzantine Empire if it was performed on Christians. So, to sum up, you could say that Jews were grudgingly tolerated in the Byzantine Empire, but they were not liked, and they were certainly not allowed to thrive.