The Fourth Crusade was staffed by Crusaders (who were overeager about how many other crusaders there would be so they asked Venice to make many more ships than they actually needed, leading to the indebtedness that Venice leveraged to use the Crusaders to depose the Emperor in Constantinople). Venice certainly supplied some of their own soldiers, but a smaller number.

The galleys that the Venetians used to fight against the Ottomans were rowed by prisoners, slaves (Venice was big in the slave trade; especially from the Dalmatian coast), captured Muslims, people with large debts. As the war between them heated up, more and more slaves were needed and raids were made on North African cities to capture men to row. The Ottomans, too, relied on slaves; often captured by 'Barbary Pirates' such as Hayreddin Barbarossa and his brother Oruc -- the former of which became the Sultan's Admiral in the fight against the Venetians.

Additionally, Venice was not just Venice -- she had territories in her Stato del Mar -- Cyprus, Croatia, Crete, Albania -- where she could pull manpower. Additionally, Venice held much of the Veneto north of her (such as my ancestral city of Padua).

So while Venice couldn't directly match the manpower of larger states, she had access to lots of different means of filling her army and her navy. Not to mention her coffers could entice Condottiere from the mainland.

For large wars like the Fourth Crusade, Venice's position as a city and a few client cities did make things difficult. They had agreed to transport 33,500 men and 4,500 horses to Egypt for the Crusade, in exchange for a large payment on a per capita basis for each man, horse, and knight transported. By Venice's own calculations, the fleet required for the transport would be 450 large transport ships and 50 war galleys, as well as all the provisions for the crusaders. Getting the provisions themselves were an issue on its own for a farm-less city. And though they didn't have to build every ship (the doge recalled the entire merchant fleet to serve as transports), the construction was Venice's largest ever building undertaking, and they still needed 30,000 sailors to man it, difficult in a city of ~100,000. If they drafted every single able-bodied man in the city they could fill it, but it would leave the city empty and without protection. Thus they decided to take half of their men, and held lotteries in churches of each parish. Once the priests summoned their men, wax balls with half filled with parchment were placed in large bowls and blessed by the priests. If it was parchment-free, you could stay home. Venice recruited the other 15,000 then from dependencies on the mainland and Dalmatian Coast.

In other wars like later mainland fights on the Italian mainland, Venice heavily used mercenaries like other Italian city states. And in times of major crisis, like the Genoese invasion in 1378, the state drafted as much of the common folk as they could and quickly trained them in the rowing and manning galleys for defense.

Unfortunately, that's as far as I reached in the book, but hopefully that's illustrative of some of what you're asking. I would definitely recommend the book as well.

Source:

Venice, A New History by Thomas F. Madden