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While the earliest evidence in texts written in the Latin alphabet for the sound change of Latin [w] to [β] or [v] may date to the first century AD (as discussed in Nathaniel's answer), evidence in the form of Greek-alphabet transcriptions of Latin V with the Greek letter β (instead of the digraph ου) seems to show up considerably earlier.

While the earliest evidence in texts written in the Latin alphabet for the sound change of Latin [w] to [β] or [v] (which can be referred to with the linguistic term "fortition", i.e. "strengthening") may date to the first century AD (as discussed in Nathaniel's answer), evidence in the form of Greek-alphabet transcriptions of Latin V with the Greek letter β (instead of the digraph ου) seems to show up considerably earlier.

I'm skeptical by default of the suggestion in Nathaniel's answer and some of the sources cited there that the phonetic change (as opposed to orthographic confusion) "[made] inroads first in the lower classes": the fact that C Novius Eunus made spelling mistakes of B for V that were avoided by an educated scribe of his own era could just mean that the scribe was more familiar with conservative spelling conventions, while not necessarily using a pronunciation that was any more phonetically conservative than that of Eunus. I am not aware of any Latin authors that comment on the pronunciation of V as a sign of social class distinctions (whereas we do find such comments about the pronunciation of ae, au, or h).

I'm skeptical by default of the suggestion in Nathaniel's answer and some of the sources cited there that the phonetic change (as opposed to orthographic confusion resulting from the change) "[made] inroads first in the lower classes": the fact that C Novius Eunus made spelling mistakes of B for V that were avoided by an educated scribe of his own era could just mean that the scribe was more familiar with conservative spelling conventions, while not necessarily using a pronunciation that was any more phonetically conservative than that of Eunus. I am not aware of any Latin authors that comment on the pronunciation of V as a sign of social class distinctions (whereas we do find such comments about the pronunciation of ae, au, or h).

The change possibly progressed at different rates in different phonetic environments

The change may have progressed faster in certain phonetic environments