Yes, during Jesus' lifetime, coins in the Roman Empire bore the likeness of the Emperor - who happened to be Augustus, also known as Tiberius, for most of Jesus' life.
These are actual Roman coins from Jesus' lifetime, bearing the image of Augustus Cæsar.
Contrary to popular belief, coins bearing the image of a pagan were not inherently considered contaminated or unfit for use in the Temple:
When Jews traveled to Jerusalem from other lands, they brought money for room, board, and souvenirs. Most importantly, they were required to pay the annual half-shekel tribute to the temple. The currency they had would be of their native land or acquired in trade along their way.
Money changers performed a key service when they converted the varieties of local coinage into the required tribute of silver shekels or half-shekels of Tyre (Tosefta Ketubbot 13:20, Exod 30:11-16). Many writers have suggested that the Tyre currency was preferred because it did not defy the Decalogue by depicting the graven image of a foreign king, and that is true. But the Tyre coins portray a pagan god of Tyre, Melqarth-Herakles—which was certainly even more offensive!
Images on coins, however, do not contaminate them even for payment to the temple. The Mishnah explains that money is unclean only if it is used for another purpose, such as for jewelry (Mishnah Kelim 12:7). The law stated that the temple must not be shortchanged in any way, so the silver coins of Tyre were most likely mandated because they were of good silver and true weight at a time when many coins were debased or lightweight.
Coin used at the Temple: