This is a complicated question which I'll do my best to answer. The TL;DR version is: they mostly co-operated until the end of the 12th century, but it was always a problematic alliance.
-Origins of the Crusades- You might know that the First Crusade found it's origin in a call for aid from the Byzantine Emperor to the Pope. In the last half of the 11th century, the Byzantines sufferd some staggering defeats in the east of Anatolia on the hands of Seljuk Turks, eventually losing control over most of the peninsula. This loss that was nothing short of a disaster for the Byzantine state; Anatolia had been their main pool of resources, human as well as natural, and there was a not wholly unrealistic fear that the Turks would soon set up camp in front of Constantinople, the Queen of Cities. It's within this context of heavy military pressure and fear of further invasion that Emperor Alexios I appealed for Western/Latin aid to repel the invaders from Anatolia, a call that Pope Urban II enthousiastically answered for a multitude of reasons that I’m not going to describe here.
-East and West in interaction- Now, the relationship between Byzantium and the West had always been a complicated one. While many Europeans looked up to the great riches and civilization of Constantinople, Byzantines were often described as effeminate and decadent and, above all, not to be trusted on their word. Vice-versa the Byzantines gave little attention towards the ‘tribes of the West’; texts by educated Byzantines like Nicetas Choniatas and Anna Komnenos illustrate this in their general indifference towards the ‘Germans’ and ‘Celts’, who were stereotyped as unruly, violent and greedy. After 1071, the West was arguably slowly overtaking the Byzantines in military and economic power. The 11th century is also the century of the ‘Great Schism’ between the Orthodox and Catholic churches, even though I reject the idea of a sudden divergence between the churches. But that’s another story for another question. The most important I’m arguing here is that, even though there was much talk of ‘one christendom’, Byzantium and Europe were by this point already clearly divided by many customs, cultural as well as religious, and the ‘alliance’ between the Crusaders and Byzantines was always a marriage of convenience, not a marriage of love. The Crusaders vowed to take Muslim territory and return it to the Byzantines, while the Emperor pledged to support, supply and transport the Crusaders when needed. On paper it looked like a great deal.
-First Contact- When the first crusaders arrived at Constantinople in 1096, Emperor Alexios was heavily dissapointed. While the Emperor had envisioned a group of highly trained and pious soldiers, what he got was the destitute, untrained and poorly armed rabble of the so called ‘People’s Crusade’; 30.000 European peasants who were uncontrollably ravaging the countryside around the city if the Emperor did not provide them food. As Princess Anna Komnenos puts it; “For the Frankish race, as one may conjecture, is always very hot-headed and eager, but when once it has espoused a cause, it is uncontrollable.” Not sure how to deal with this enormous crowd, Alexios Komnenos decided to ship them to Anatolia as quickly as possible, where the People’s Crusade found a bloody end on Seljuk swords, with only a fraction escaping to Constantinople. I believe this episode is somewhat exemplary of the whole Byzantine-Crusader interaction; differing expectations, cultural barriers and miscommunication abound. They nominally worked together, but always as two disctinct and often rivaling groups.
-Disagreements- As the crusading age went on, friction between the to groups was rising as well. As an example: Crusaders blamed the Byzantine Army for not helping them out during the Siege of Antioch in 1097. According to some Crusaders the Byzantines had kept their army at bay on purpose, hoping the Crusaders would all die at Turkish hands. Vice-versa, the Byzantines were shocked to hear that the Crusaders were not planning to return Antioch at all, but instead make it a Latin principality under Catholic jurisdictions. Both sides repeatedly broke their promises and would do so many times aftwards, leading to a kind of ‘he-say she-say’.
-"'1204' and all that"- These kinds of conflicts were a recurring element of the Latin-Byzantine relationship during the 12th century, and wre slowly setting the stage for the great plundering of Constantinople in 1204, during the Fourth Crusade. This attack on Constantinople can be seen as a sort of final split between the Crusaders and Byzantines, who afterwards never entered a equal alliance again. At the end of the 12th century, the Byzantine Empire was in a very bad state, and the Latin west had definetly overtaken them in terms of military power. The leadership of the Fourth Crusade decided to lay siege on the city of Constantinople to help the son of a deposed emperor take back his throne. They did this succesfully and without too much violence, but after a palace coup brought an anti-crusader emperor to the throne, the city was attacked and brutally sacked by the Crusaders. Large parts of the Byzantine Empire were now subjugated by these Crusaders, and Constantinople was made the capital of a new, ‘Latin’ Empire under the Pope’s authority. After the establisment of the Latin Empire, the Crusaders and remnants of Byzantium perpetually fought eachother until the recapture of Constantinople by Greeks in 1261.
-Conclusion- So, to make a long story short, while the Byzantines and Crusaders were nominally allied against the muslims and did indeed support eachother and battled alongside eachother, the relation was always a difficult one as both sides had considerably different interests. The understanding between the two groups slowly deteriorated over time, and found it’s climax in the Sack of Constantinople of 1204.