The US doctrine of containment (and in general most of the interventionist doctrines) in the Cold War didn't simply demand a military presence in potential invasion zones. A powerful navy, historically, has been used as far more than simple military deterrence. From the days of the Royal Navy, and the US's first naval decades (the 1908 Great White Fleet tour), a powerful navy is used as a demonstration of power projection and reach. The United States continues to do this today, with powerful fleets patrolling and exercising in areas that are not currently conflict zones. The US were not the only ones to do this; during the Cold War era the Soviet Union invested heavily in its navy to carry out the same purpose, including in the Black Sea and the Mediterranean. "Presence" is more than just having a floating airfield; after all, American planes could launch from NATO airbases all over the region. "Presence" is the control of shipping lanes and the ability to land larger scale land units (as opposed to smaller, lighter airborne forces).

However, in the case of the Mediterranean presence, there were many real strategic objectives. Early in the Cold War, the Soviet Union pressured Turkey, and in general, the states on its southern border, as such states like Greece, Turkey, and Iran were some of the remaining unaligned states. The buildup of naval forces in the region during this period combined with a diplomatic effort to reach out to Greece and Turkey, both of which joined NATO in 1951. Incidentally, this meant that the Soviet Black Sea Fleet was essentially sealed, and posed no threat to the Mediterranean, eliminating the only real "doomsday scenario" (event of war with the USSR) target.

However, not the entire Mediterranean fell under NATO's sphere. With the Western sponsorship of Israel, the USSR found easy friends in the Arab world, Syria and Egypt. The USSR leased naval bases from both these nations (Tartus is still leased to this day), and a naval presence returned to the Mediterranean.

Strategic control of the Mediterranean is extremely important to any campaigns in the whole region. First of all, the Suez Canal's importance speaks for itself. Second of all, control of the Mediterranean means striking capability anywhere along the shore. In one of the first written treatises on naval power projection, Mahan breaks down how Rome defeated Carthage -- control of the Mediterranean. The same strategic importance was displayed again in World War 2, where Axis supply efforts struggled because they could not achieve naval or air superiority in the Mediterranean (the British went around Africa). I don't know of any specific war plans or of the role of the Sixth Fleet in the case of a full-scale war in Germany, but in the event of war, Greece and Turkey would definitely require support, in the form of arms, supplies, and men that had to go through the Mediterranean.

As far as actual action, there was some. The navy participated in the intervention in Lebanon in 1958, as well as non-violent confrontations with the Soviet navy during the Yom Kippur War when the Soviets threatened intervention. The Sixth Fleet has since been scaled down, but in the Mediterranean, as in a great portion of the rest of the world's waters, the US Navy keeps a nearby presence to display American power projection, whether that means threat of violence, denial of hostile passage, protecting American strategic interests, or even serving in emergency civilian capacities like searching for missing planes, delivering aid (like Yugoslavia in the 90s), or, more recently, defending against pirates.