She was a fairly regular adulterer, and so her 'crime' was adultery, though she was also charged with treason. I'll get to that. Firstly however, regarding adultery, the Late Republican Romans had a slightly loose view of adultery in general by our standards, free as they were from Christian morality at this time, but by no means was such behaviour considered a good thing. Just as qualities such as virtus and gravitas were considered essential to the reputation of a man, modestia and pudicitia (modesty and chastity) were essential values for a Roman matron. The archetypal Roman matrons were women like Cornelia, the mother of the Gracchi and Aurelia, the mother of Caesar. These women dedicated themselves to child raising and after the death of their husbands, lived chaste and family-oriented lives.

On the other hand, Roman marriage for the nobility came in several different forms, but they were generally marked for their political and economic advantage rather than anything to do with love. Marrying for love was an unusual thing, perhaps even deviant, and showing too much affection to ones spouse in public was considered strange. These upper class women were also ladies of leisure and had a great deal of time to themselves while their husbands engaged in politics, education and networking. As you can imagine, these bored women in loveless marriages in an environment where men often had sex with household slaves, might have lovers. In fact, this does not seem to have been unusual.

So the issue with Julia, the daughter of Augustus, is that Augustus was engaged in a program to restore the Republics ancient morals, which included legislation against adultery. Aside from the fact that Augustus first began a relationship with his longterm spouse Livia while she was still married, this was highly embarrassing for Augustus, the self proclaimed exemplar of old Roman morals. He had actually passed legislation against adultery that could see women deprived of dowries and even allow cuckolded men to kill their wife's lovers. And Julia conducted affairs for years with multiple husbands to the point where it was basically public knowledge. It was the fact it was so well known that was the key problem. The final straw was that some of the men Julia engaged in relationships with were also implicated in plots against Augustus and Tiberius, the latter of whom was Julia's (estranged) husband at the time. The details are murky, but such a conspiracy would not have been unusual, though I do not know the extent of her personal involvement. Considering how long she had been behaving like this, a threat to Augustus' arrangements for the succession is one thing that Augustus could not have compromised on, she was too much of an embarrassment and political liability to her father, who appears to have been very patient with her affairs for the most part, and he was forced to act against her.

While some may have thought Augustus was harsh in private, such authoritarian behaviour toward ones children was at least in keeping with old Republican values. Julia in many ways appears to have been quite likeable, but in the end a threat to Augustus' plan for succession, real or perceived, could not be allowed to stand and I think most upper class people would have understood this. The Roman people might have missed her as, for what it is worth, Macrobius says she was popular, but they loved Augustus a lot more and would have respected his authority as paterfamilias, not to mention princeps.

Sources:

Eck, Werner. The Age of Augustus

Scullard, H.H. From the Gracchi to Nero: A History of Rome from 133 to 68 B.C.