The problem starts with Archipelago maps: sure, it's fun to go island hopping, but Archipelago maps are generally a worse experience in Civ5 than most other map types. Unlike land warfare, naval warfare is not really an option for most Civs until Navigation; maybe you can get something interesting going with Galleases if have nearby opponents, but otherwise this means that you'll pretty much have no way to take anyone's cities until Renaissance. Next, naval warfare in general is a lot faster and frankly, a lot more boring than land warfare until subs and carriers become available: thanks to no defensive terrain in ocean, no ships that can move after attacking, and no healing in non-friendly territory without a special promotion, naval melee units are pretty much useless for anything other than capturing cities (and capturing almost-dead ships as Privateers). In addition, the high base movement speed for ships (5, as opposed to 2 for land units) coupled with the fact that ships cannot get any defensive bonuses from either terrain or promotions means that the person who attacks first with their ranged ships will have a huge upper hand, even more so than first move ranged attacks on land. Naval warfare in Civ5 is like being forced to fight all your wars with Crossbowmen until Modern Era, including any Classical Era wars. This also ties into my next point quite well, in that as a result of certain naval units having such a dominant role, civs whose UU's replace those dominant units end up being so incredibly overpowered that it's just not funny (England). Great Admirals are much less useful than Great Generals because Great Admirals have 3 less movement speed than the average naval unit, so by the time they catch up to your navy fighting a fight, the outcome will have already been decided, making your Great Admiral irrelevant. Finally, water tiles in Civ5 have a much lower average yield than land tiles, so games played on maps with primary water tiles will tend to be much slower due to the lower yields your cities will be generating.
And let's not even mention the absolutely terrible naval AI.
Now let's get to Carthage itself. Free harbors are a nice UA, but they really pennies compared to the bonuses other civs might get on archipelago maps, be it instant, free luxuries (Indonesia), land units that don't instantly end their turn after embarking or disembarking (Denmark), or a +2 movement boost to all naval units (England). Even certain civs who aren't specifically naval, eg. Poland, Egypt, Japan, could be considered to have a better UA for naval maps than Carthage's free Harbors. If we start taking UU's and UB's into account, Carthage is beat as well. Carthage's only other naval boost is their Trireme UU, which is absolutely terrible: +3 strength is useless when it's only on a melee ship that has less utility than a Spearman (costs more hammers, has less strength, can't use to clear barb camps or to defend settlers from being captured by land barbarians, can't use to defend vital tiles because of no defensive bonuses). By contrast, Byzantine's Trireme UU is actually useful because it's like a naval Chariot Archer, so it can actually be used to swarm and take cities, and that's before you consider that Byzantines' bonus belief can also be used to get beliefs that are more advantageous on naval maps than the free harbors. England's Ship-of-the-Lines are monsters, as they can only really be countered by submarines. Even Venice's lowly Great Galleases, which only differ by a measly +2 melee strength and +3 ranged strength compared to Galleases, have more utility than Quinqueremes simply because Galleases are ranged naval units, while Quinqueremes are not.
Basically, Carthage is not the worst civ for naval maps, but there are still so many better options, including some who are just as good on naval maps as they are on land-based maps (eg. Poland, Egypt, Maya, Byzantines, England).
Now let's get to your actual strategy.
Free harbors are pennies when you're going Tradition: because they are not food, hammer, culture, or science buildings, they are worth the most when you have a lot of cities that need Harbors. You'll get a lot more use out of free harbors if you have 8-10 cities: the less gold from city connections will be made up for by not having to pay that 3 gold maintenance, and not having to spend 120 hammers on harbors means a lot more when your cities wouldn't have the hammers to build them anyway. Plus, naval maps tend to be lower on food anyway thanks to not having as much farmland, so you probably won't be able to get your Tradition to as high of a population as you want anyway (unless you're Aztec because Floating Gardens).
Since trade routes are player-capped in Civ5, there's not much use building harbors for the trade route gold in more than one city, so free harbors are needed only for the city connections and Seaports after Navigation. Colossus is a good wonder regardless of Harbors: that +1 trade route is incredibly useful.
