In the 1st century, Judea was a Roman province. There were actually a wide range of systems in use in Judea in the first century AD, but the Graeco-Roman system was the one available to everyone in the province.
Roman law was quite explicit in permitting women to swear oaths and testify in court.
The Digest of Justinian states:
The fact that the Lex Julia [legislation introduced by Augustus in 23 BC] on adultery forbids a woman found guilty to give evidence shows that women have the right to give evidence at a trial.
A number of other references to women being able to appear in court as witnesses appear in Book 12 of the Digest of Justinian.
[Although the Digest of Justinian is a compilation that was put together much later (in the 6th century), the laws were often enacted much earlier - as in the case of the Lex Julia mentioned above].
Furthermore, in his trial against Verres in 70 BC, Cicero called several women as witnesses. In his speech, he shamed Verres for having forced him to compel respectable women to appear in court to testify against him. Cicero's objection here is clearly against disturbing women of station, and not against trusting their testimony, otherwise why would he have called them as witnesses?
Indeed, we have a number of examples from ancient Egypt (another Roman province, but one where we generally have more surviving epigraphic evidence from the period) of women being allowed to testify in court:
One example (P.Oxy. 1.37) dated to 49 AD, which was copied from an official government archive, shows that a woman's testimony was entered into the court record.
Another example (BGU 4.1105), from a case in Alexandria in 10 BC, records the testimony of a woman who testified in court against her husband for divorce on a charge of wife-beating and squandering her dowry.
(The translations appear in Greek, but Google Translate should be able to make a decent fist of translating them into English. I'll try to locate English versions & update the answer later, when I'm not typing on a phone screen.)