We should first make a distinction between two types of crusader. The first we can describe as those journeying from Europe to participate in certain campaigns for a certain duration. The second are those families who remained in the Levant for the creation of the crusader states and subsequently became the local landed elite. Their approaches in communication will have been different.
When the First Crusade left Europe few of its participants will have had a command of Greek and/or Arabic. This meant the leadership was 'relying primarily on outsiders to act as intermediaries' (Clason, p.338). This could be in the form of local interpreters, such as Byzantines or Syrians, or they could be specifically attached with the crusader army. Bohemond I of Antioch had his own translators for both Greek and Arabic and his ability to negotiate with the local population was key in his success at the siege of Antioch in 1097-98.
In this way, 'outsiders' to the region would employ people when negotiations had to be conducted in Greek or Arabic, just as staff were employed to fulfill any number of roles. However, communication subsequently became more intimate as the Europeans entrenched themselves within Palestine.
The people we refer to here are the 'second-generation Eastern-born nobles'. Clason writes that 'knowledge of the Arabic language, with its relationship with Muslims and strong Eastern identity, reached the highest echelons of Latin society by the final quarter of the 12th century' (Clason, p.338). Humprhey II of Toron, constable of Jerusalem, spoke Arabic, as did Reginald, lord of Sidon. The one time regent of Jerusalem, Raymond III of Tripoli was himself an Arabic speaker. 'The royal connections seen here also suggest that more Latin barons could have acquired such language skills'. In this way, the second generation crusaders had a very different relationship with the local language than their predecessors.
The crusaders could choose to employ the clergy, such as in the case of William of Tyre. We can't assert any level of fluency in William's Eastern languages, but his work suggests at least a cursory knowledge of Greek and Arabic. He represented king Amalric I in an embassy to Constantinople in 1168. Likewise, the Holy Orders will have had brothers with knowledge of local language. When the delegation of Hugh, lord of Caesarea met the Egyptian caliph in 1167, he took with him a Templar by the name of Geoffrey, because he 'probably spoke Arabic' (Runciman, p.373).
These are some of the strategies that the crusaders used when communication needed to be conducted in Greek or Arabic, while between themselves they largely used the courtly French familiar in Europe. The original crusaders and subsequent adventurers relied on local translators or else arranged for their own to accompany them. Later, the regime could use expertise from the clergy and Orders, where it was available. Finally, as the nobility became Eastern-born, more and more will have been able to conduct their own negotiations in Arabic.
Hope this helped a bit!
Read more:
C. R. Clason, ch.7 in East Meets West in the Middle Ages and Early Modern times, Albrecht Classen (2013).
C. Tyerman, God's War (2006).
S. Runciman, A History of the Crusades Vol.2 (1952).