Yes, the Cossacks went on naval raids all the time - it was a major realm of Cossack activity.

The main authority on this is Victor Ostapchuk, who has investigated the Cossack presence in the Ottoman-controlled Black Sea during the seventeenth century. I was also talking with him just this morning, so your question comes at a coincidentally good time. The important article here is "The Human Landscape of the Ottoman Black Sea in the Face of the Cossack Naval Raids" (2001), which covers this topic in some detail.

The Ottoman Empire of the mid-16th Century controlled every port on the coasts of the Black Sea and had effectively stamped out piracy. Ostapchuk describes it as an "Ottoman Lake". Not having faced any military threat there within living memory, the Ottomans were generally unprepared for the appearance of the Cossacks in the second half of the century. Operating out of long boats called chaikas in Ukrainian, they sailed down the Dnieper to attack Ottoman fortresses on the north shore of the Black Sea, gradually expanding their activity to raids on the Crimea, and the coast of Ottoman Europe down to the Danube and Bulgaria. The big shock came in 1614, when they suddenly crossed the open sea to destroy the Anatolian town of Sinop. Per the chronicle of Hasanbegzade, quoted by Ostapchuk:

...going over the top of its [fortress'] ramparts and walls they entered inside and descended upon the center of the city and destroyed its circumference and edifices and shed the blood of several thousand men and woman and struck the mentioned city with the broom of plunder and the fire of devastation and they left neither name or nor sign of its buildings, turning it into a wilderness and a desert.

Thereafter the Cossacks looted innumerable Ottoman cities, essentially every place they could reach either on the coast or slightly inland. They operated large fleets with hundreds of ships and thousands of men, taking slaves and booty wherever they went. The Ottomans responded by establishing naval squadrons for patrol and by tightening their control over the mouth of the Dnieper, forcing the Cossacks to raid in smaller numbers, but they were never able to totally prevent them from entering the sea. After Sinop, the Cossacks attacked Kaffa (1616), Varna (1620), and Trabzon (1625), to name just a few, and often revisited sites earlier destroyed. Most famously, In 1624 they even made their way up the Bosporus to loot the suburbs of Istanbul. Trade on the Black Sea became a dangerous business, as well as trade along waterways connected to it. Cossack bases located in secluded stretches of the Danube delta meant that even this vital lifeline of the empire became unsafe. Although often romanticized in Eastern Europe as religious warriors, they seem to have had no qualms about looting Orthodox Christian populations as well as Muslims. Ottoman sources describe the Cossacks as a terrible scourge and relish every naval victory of the Ottoman patrols whenever they could catch a Cossack fleet off-guard. Curses are showered upon them whenever they are referred to. Ostapchuk argues that their activity terrified the Ottomans and significantly changed their relationship with the Black Sea, which was transformed from a safe haven into a place of foreboding and danger, as evident in the writings of the traveler Evliya Çelebi, who "at times seems almost haunted by Cossacks."

By the 1620s Cossack raiding against the Ottomans reached such an extent that it led to a breakdown of Ottoman-Commonwealth relations. In 1621 Sultan Osman II led an Ottoman army to Hotin, one of the goals of which was to force the Commonwealth to restrain Cossack raiding. His effort was a failure, and only exacerbated the problem as the Commonwealth had allowed Cossack ranks to swell in exchange for their participation in the campaign. In 1636 the Cossacks even went so far as to occupy the Ottoman fortress of Azov, which they held for several years, and to pledge the fortress to Muscovy (they turned down the offer). The real turning point was the revolt of Khmelnytsky in 1648, which finally drew Cossack attention away from the Ottomans. Thereafter Cossack leadership was forced to maneuver between the various powers active in Eastern Europe, including the Ottomans, which encouraged them to maintain as positive of relations as were possible. Incidentally, this applies solely to the Zaporozhian Cossacks - the Don Cossacks continued their naval raiding throughout the second half of the century. In 1669 Cossack Hetman Petro Doroshenko offered his vassalage to the Ottomans. They accepted and in 1672 invaded the Commonwealth again in order to protect their new vassal and shore up their own control over southern Ukraine, annexing the province of Podolia with its fortress of Kamianets-Podilskyi (Kamaniçe). They remembered the devastation which the Cossacks had inflicted upon the Black Sea and resolved to keep that frontier safe. Ottoman rule lasted until 1699, when the Treaty of Karlowitz signed in the aftermath of fifteen years of warfare with the Habsburg Monarchy, Polish-Lithuanian Commonwealth, Venice, and Muscovy stipulated that the region should pass back into Commonwealth control.