I'm not well versed on their reaction to all of the Crusades, but their opinion of the First Crusade was initially very positive. Emperor Alexius had extracted oaths of loyalty from the prominent Crusader leaders, and promises to restore reconquered territory in Anatolia to the Romans, and this is what happened initially. Alexius gave the Crusader armies extensive supplies for their journey, as well as knowledgeable guides. He sent his navy along the coasts to help them whenever possible, and eventually sent armies in their wake to protect the conquests of the Crusaders (which, naturally, he wanted to see restored to his empire). Cooperation between the Romans and the Crusaders was effective and relation were relatively positive - both parties had the same objectives; to defeat the Seljuks and restore land to the Christians. However, things took a sharp turn when the Crusaders refused to hand over Antioch to the Romans following its capture in 1098. Alexius was understandably angry, since it was in direct violation of their previous agreements.
Things took a further negative turn when the Crusaders revealed their intention of invading the Fatamid Caliphate and retaking the Holy Land. Alexius was on good terms with the Fatamids, since both of them viewed the Seljuks as a more serious threat than each other, and they agreed that it was in both of their best interests to focus their attention on defending against the Seljuks rather than fighting each other. Alexius warned the Crusaders not to start a war with the Fatamids, but they ignored him and marched down the Levant coast. The Emperor was furious and promptly cut off all aid he was giving them, including supplies and naval assistance. For the rest of the First Crusade (which at that point continued for only a few more months) there was no cooperation between the Romans and Crusaders. Alexius saw the invasion of the Holy Land as pointless from a strategic perspective and in nobody's best interests, and quickly distanced himself from the Crusaders and tried to maintain positive relations with the Fatamids, telling them he had nothing to do with it.
The only Crusader state that the Romans had somewhat positive (or at least not overtly negative) relations with was the Principality of Antioch, since its prince, Bohemund, swore fealty to Alexius as his overlord. This took some of the sting off the Crusaders' refusal to hand over the city, although Roman control of the principality was minimal and tensions were always high, since the princes simultaneously were influenced by the Kingdom of Jerusalem as well.
Source: God's War: A New History of the Crusades by Christopher Tyreman
u/Wonderfully_Mediocre’s not at all mediocre post already goes over the First Crusade, so I’ll just link to some previous answers of mine about that: When the Crusaders left Anatolia, was the Levant still covered by their oath to return territory to the Byzantine Empire? and Why did Bohemund declare himself the Prince of Antioch, not the King?
After the First Crusade, the Byzantines were happy that Jerusalem and the other holy sites were in Christian hands again, but they wanted to make sure that the rights of Greek Christians were respected (which they were very often not, under crusader rule). Otherwise they didn’t really think anything of it from a political point of view, since Jerusalem was strategically unimportant to them. They did have a fairly good relationship with the crusader kingdom in Jerusalem; the royal families intermarried, and they tried to conduct military expeditions together (particularly against Egypt), but the Byzantines didn’t really want to rule anything except Antioch.
For the Second Crusade, both Louis VII of France and Conrad III of Germany travelled through Constantinople, as the First Crusade had done. In that case, the Byzantines were fairly worried:
“The appearance of a large crusading army was a cause for grave concern in Constantinople. Unlike in the background to the First Crusade, when Alexius I had requested a western force to come to his aid, there was no such invitation in 1146-7.” (Phillips, The Second Crusade, pg. 170)
Sicily had been attacking Byzantine territories around the same time, so the Byzantines worried that the crusaders were working with the Sicilians. They were also worried that the crusade would besiege Constantinople, but the crusaders had no intention of attacking the empire and they were eventually ferried across the Bosporus and made their way to Jerusalem.
Conrad wasn’t technically the Holy Roman Emperor, so there was no problem with two emperors both claiming to be the Roman Emperor, as there was on the Third Crusade. I answered a previous question about that as well: Holy Roman Emperor Frederick Barbarossa and Byzantine Emperor Isaac II both claimed to be the legimitate Roman emperor so it was extremely difficult for them to agree to anything. The other problem for the Third Crusaders was that the crusaders believed the Byzantines were secretly allying with Saladin to destroy them. The Byzantines were probably just exploring all their diplomatic options though, not conspiring against the crusade specifically.
In any case, the problems on the Third Crusade were also overcome and the German crusaders made their way to the Holy Land (although Emperor Frederick died along the way). The other contingents from England and France avoided the overland route and took the sea route, but the English managed to somewhat accidentally conquer the Byzantine province of Cyprus. The Byzantines were not particularly happy about that, but Cyprus was ruled by a rebellious lord, so it was really already outside of the authority of Constantinople.
The Fourth Crusade is of course an entirely different matter…the Byzantines, in hindsight, obviously felt that it was a huge disaster for them. Very briefly, the emperor at the time, Alexios III, had deposed the previous emperor, his brother, the aforementioned Isaac II. Isaac’s son Alexios IV came to an agreement with the leaders of the crusade to divert it from its original target (Egypt) to Constantinople to restore Isaac to the throne. So, there were at least some Byzantines who were in favour of that…but not many, since the people actually living in Constantinople were opposed. Alexios IV and the crusaders did manage to restore Isaac to the throne, with Alexios IV as co-emperor, but then Isaac died. Alexios IV was then overthrown and murdered by yet another Alexios, who became Alexios V. The crusaders were still there though so they attacked and conquered the entire city. The disastrous part of this in Byzantine eyes was not the rapid overthrow and murder of various emperors (which happened often enough), but the crusader sack of the city. The Byzantines were always paranoid that a crusader army showing up at their door would sack the city, and finally with the Fourth Crusade, it actually happened.
The other crusades then largely bypassed Constantinople, because it was already under Christian control - interestingly, the problems faced by the First, Second, and Third Crusades, arriving at a somewhat hostile Constantinople, would have no longer been problems for the Fifth, Sixth, and Seventh Crusades, but the later crusades avoided it anyway and took the faster sea route.
Sources:
Charles M. Brand, Byzantium Confronts the West (Harvard University Press, 1968)
Jonathan Harris, Byzantium and the Crusades (Hambledon and London, 2003)
Jonathan Phillips, The Fourth Crusade and the Sack of Constantinople (Pimlico, 2005)
Jonathan Phillips, The Second Crusade: Extending the Frontiers of Christendom (Yale University Press, 2007)