You could—the letters are substantially the same (functionally speaking) as the modern Hebrew alphabet—but it is graphically less sophisticated than its derivatives, namely the Greek & Roman alphabets, and the Assyrian script which gave rise to the revised form of Hebrew from the Second Temple period onwards, known today as “Square Script”:
Compare the Phoenician/Old Hebrew:
with conventional, Square Hebrew script of today:
The old script lacks the graphic consistency, rhythm, or concision of the Square Hebrew.
However, if you insisted on using graphic forms that were similar to those of Old Hebrew, you could use a variation of SimHebrew, which simulates Hebrew spelling in Roman characters:
which reflects some similarity to the Phoenician, when flipped horizontally, as the ancient Greeks eventually did: