History is full of a large number of incredibly tantalizing 'what-if' scenarios. It's impossible with any real accuracy to predict what would happen in Barbarossa's army hadn't met with tragedy at Calycadnus. But, I'll try to make at least some guess in order to provide a satisfactory answer.

Barbaroosa's army was the greatest ever assembled to go on Crusade. Common thought, among the Christians, was that his force was strong enough to repel the Muslims alone. Combine this with Richard's English army and Philip's French, and it is certainly very possible to entertain the idea of retaking the Holy Land.

Saladin actually assembled his forces to meet the incoming Germans. Working with the Caliph, Saladin had assembled his forces to attempt to hold the Germans in Northern Syria. When news came of Barbarossa's death and the splitting of his army, these troops could be safely sent to other parts of the Caliphate. Not only that, but the soldiers of Islam themselves saw it as a miracle from God and a sign of their righteousness.

Without the Emperor, the German troops had no real leader and once again split into squabbling princes (As the Holy Roman Empire seems to quite enjoy doing).

All this makes a very convincing argument towards the success of the Crusade, and it is certainly possible to make that guess. Though, if Barbarossa had made it to the Holy Land--we have to wonder if Richard would have a chance to match tactical wits against Saladin as he did. We have to wonder about the fighting between Germans, French, and English (Which, even without Barbarossa, would lead to the Duke of Austria growing so infuriated with Richard--he would head home and imprison the King of England on his return voyage some years later.)

The medieval Crusader leaders always had their own politics and agendas. They were not equipped to fight in the Holy Land on any of their long ventures, and severely misunderstood their opponents at every turn. The Frankish soldiers always believed in the valiant charge, with God as their ally. The Islamic soldiers would harass and pick off weak crusaders trying to navigate across the hot deserts of Syria and the Holy Land in full armor.

So, it really is impossible to say one way or the other. I, personally, do believe that the goal of the Third Crusade might have been achieved--though I sincerely doubt that the overall Crusades would end in any different outcome. I believe we would still see the eventual decline in the Crusading spirit, and greed could always be fulfilled a little closer to home than the Kingdom of Jerusalem.

Sources: (Runciman: Chronicles Of the Crusades Volume 3 ~page 13)