I’ve always felt like this single event may have changed the entire history of the Holy Land and subsequently that of Christianity and Islam themselves. Frederick was an establish commander and the crusader armies of Phillip and Richard would have cooperated under his command. I believe this crusader force would have been far more successful than any other had it stayed together.
What if? - questions are always difficult for historians to deal with in a professional manner but they're fascinating nonetheless. Considering the third crusade was the one with the three most powerful leaders in western Europe, King Richard of England, King Philip II. of France and Emperor Frederick I. Barbarossa of the Holy Roman Empire, it was set out to be the most successful one in terms of possible conquests. They as well as many other Christian leaders of Europe made peace and united to follow the call to a new crusade made by pope Gregory VIII and then Clemens III. after Gregory's death. So considering the participants and the preparation that went into this 3rd crusade (especially on Frederick's side) it should have been the most successful yet and managed to retake the Holy Land, Jerusalem and more. The numbers of contemporary sources vary a lot on the amount of troops and knights that went on the crusade on behalf of Frederick's call, but the number of his troops was much larger than Richard's and Philip's, estimated at about 100,000 people (not all combatants), which also had grown on his journey through eastern Europe (especially Hungarians joined in larger numbers).
Frederick being the oldest and as Emperor of the Holy Roman Empire also the most respected one of the leaders most likely would have been able to mediate the animosity between Richard of England and Philip of France as well as others and therefore could have kept the crusaders united for a longer time.
Frederick's troops were well trained and lead as victories against the attacking Byzantines under Emperor Isaac II. Angelos as well as some turkic Seljuks on the way proved.
Frederick's death in the river Saleph, in what is now southern Turkey, demoralized his troops tremendously and most of them went back home. Only a smaller contingent under his son and later under Leopold V. of Austria participated in the fights to come, but even then left after the city of Acre was taken back from Muslim troops.
Considering lastly that the 3rd crusade was in its entirety not at all a failure and Richard and Philip had made great successes against Saladin, a most accomplished and capable leader, although not retaking Jerusalem they took several coastal territories and managed to open Jerusalem for Christian pilgrims and merchants again as well as establishing a European Christian presence in the Holy Land again.
To conclude, there is enough evidence to say that Frederick's death had probably a huge impact on the overall success of the 3rd crusade and its aftermath. However if those suspected successes would have changed the course of history, e.g. the survival of the Byzantine Empire or even a Christian state in the region is of course a matter of speculation.