Galata, also known as Pera in Latin sources, was officially neutral in the siege of 1453 as it was not specifically a Greek/Byzantine settlement by that point. It was instead the suburb across the Golden Horn of the Genoese merchants with its own walls and fortifications. For this reason you are quite correct in saying that it was supposedly neutral despite its highly strategic position. The Turks had good reason to allow this, Galata and Genoese merchants continued to sell them supplies. By Venetian reckoning, the very castle of Rumelii Hisar which Mehmet built to besiege the harbour used Genoese wares.

That is not to say the Turks respected this agreement of neutrality outright and in perpetuity. During the siege Mehmet is known to have bombarded Genoese ships to force them away from the chain stretched across the harbour. They naturally complained and the sultan's position was to deride Genoese impartiality by labelling them condottieri 'mercenaries' of Konstantine and Konstantinople.

Interestingly, the Ottomans were probably correct in their presumption that some in Petra were aiding the Constantinopolitan defence. Giovanii Giustiniani was a renowned pirate and 'corsair' around the Aegean and Black Sea Caffa since the 1440s. The ships bombarded in 1453 were described similarly as corsairs in Latin sources. (Marios Philippides, Siege and Fall of Constantinople p.382-3).

However, regardless of reputation the Genoese of Petra certainly did not attack the Ottomans directly and they may even have collaborated with the Turks. The Venetian sources claim Galata sold information of a proposed attack upon the Turkish ships after they were transported over land into the Golden Horn. (Marco Barbaro, p.29). This could be simple mudslinging and scapegoating as neither were the inhabitants of Galata outright traitors to Christianity. The Galatan Genoese gave safe harbour to the retreating Giustiniani forces and hosted many individuals after the siege while they awaited ransom. Their loyalties were seemingly a mix of civic loyalty and basic pragmatism and while highly wealthy they were not a huge population in the first place so they were not a military threat.

The morning after the fall on April 29, Galata/Pera surrendered without bloodshed. Many of the Genoese who had fought with Giustiniani fled but the district and it's 'neutral' inhabitants continued their trade under the Ottomans who needed to protect the flow of wealth to repopulate the decimated Queen of Cities.