It ranged between enemies and allies, all depending on where the Emperor was, but for the most part they were allies under the Komnenoi (The dynasty of Byzantine Emperors during the relevant period) due to the mutual recognizing that they were Christian powers in the east and needed the Empire's protection. I'll focus on the reign of the Komnenoi, since that's when the Empire was strong enough to be relevant in the Levant.

Under Alexius, the state of Antioch attacked Alexius after the latter's supposed abandonment of the siege of Antioch so that the Crusaders set up their own state. As is expected, though, the little state of Antioch didn't stand a chance against the Byzantines and so Bohemond I was forced to sign the Treaty of Devol, which recognized the Crusader State as a protectorate of the Empire.^1 However, outside of this, Alexius's rule was much more concerned with the capture of more of Anatolia and the securing of the coastal cities under the forces of John Doukas, so Alexius's political-military interactions with the Outremer was relatively minor after the crusade.

Under his son, John II (Whose reign is unfortunately not very well recorded), the Crusader States generally understood that the Byzantines were the protectors of the States, especially Antioch, although this was something that could only be inconsistently enforced. John II exacted tribute from all of the northern states (Antioch, Edessa, and Tripoli) and, if he hadn't died, would have gotten to watch the Kingdom of Jerusalem piss themselves due to his desire to take an army to Jerusalem for an armed pilgrimage.^2

Under his son, Manuel, things get even more interesting (Manuel I Komnenos is a fascinating person to read about). Manuel's treatment of the Crusader States was one of domination, he truly saw himself as the rightful suzerain of the Outremer, although this is when you also see some open hostility between them. Rayland, the Duke of Antioch, claimed that Manuel had promised him a sum of money and didn't pay, so he invaded Cyprus and ransacked it along with his Armenia Minor allies, hoping that Manuel would be too busy to retaliate (Although he did a bad job at this, according to William of Tyre he sent the bodies of mangled peasants to show how much he hated Manuel). As was often the case with Manuel, however, he replied with great spirit and quickly sent a huge army to Cicilia and Antioch. When Rayland realized he was screwed, he bade for peace and after a comedic scene where Manuel supposedly gave him the cold shoulder like a feuding middle school couple, Manuel accepted the peace with the terms of submission. In one of my favorite scenes that encapsulates the Komnenoi so well, a grand entry of the city was done by Manuel where he rode on horseback while the King of Jerusalem and the Prince of Antioch were forced to walk behind him (a sign of lower standing).^3

Another event of note is that Manuel launched a large invasion of Egypt via the city of Damietta with the Crusader State of Jerusalem as allies. The armies were quite large and the Byzantines used their large navy to keep them supplied, but the two didn't trust each other and so were forced to pull out due to lack of cooperation.^4

After Manuel's death, the rapid decline of the Crusader States, coupled with the fact that other than Isaac II, the Emperors were busy killing each other to really do anything, meant that the Byzantines were much too busy to do anything relating to the Outremer. The decline of the Byzantine Empire, due to their acting as protector of the crusader states, is often seen as a reason for the failure of the Crusader States.

So, generally they were allies with recognition that the Byzantines were much more powerful and as such were the overlords, but with the States mostly acting independent due to the remoteness of their location compared to Constantinople and with some hostility mixed in.

Hope this answers!

[1]Norwich, John J, Byzantium: The Decline and Fall.

[2]Runciman, Steven, A History of the Crusades, Vol. II: The Kingdom of Jerusalem (I haven't quite read all of this book or it's Volume I, but I would say that this is probably a good text for what you're looking for)

[3]William of Tyre, A History of Deeds Done Beyond the Sea (Also a very good book for what you're looking for if you can find a good translation of it)

[4]Magdalino, Paul, The Empire of Manuel I Komnenos

Also, if you would like for me to give you a detailed response for how the Byzantines and the Crusader ARMIES (So, before the scope of when you asked this question), I'd be happy to :)

Edit: While looking for more information about the relationship between Alexius and the Crusader States, I found this book which might be of interest to you! http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780198204077.001.0001/acprof-9780198204077-chapter-2