don't know much about Phoenecian 'navy' activities but with their fleets and sailors they must have had the material. I'm sure they had anti-pirate forces.
I read somewhere the Romans found a shipwrecked Carthage ship and patterned theirs after. their tactics usually involved getting close to enemy ships and jumping over with sword in hand to settle things land combat way.
eventually the Roman fleet ruled the Med, supplying troops, escorting grain convoys and chasing pirates.
(I guess Carthage was an extension of the Poenecians?)
That depends on what you mean by "best"? Do you mean the best technology? The largest fleet? Most dominant fleet in it's own time? Who contributed most to understanding of naval architecture and technology and improved understanding of naval mechanics?
Be more specific than the term "best" because the "best" is whatever favorite ancient super power dominated the sea you personally like best.
If you go by the most amount of sea ruled uncontested for the longest, then surely it has to be the Romans? From the 1st punic onwards they simply dominated all mediterranean naval activity. So 600 years if you only count unified Rome, and double that if you include Byzantine.