Cultural/knowledge transfer is famously difficult to prove unless it is closely documented. Unless the person specifically doing the transferring wrote down at some point that they had specifically gotten the idea for this in some other place the best you have is circumstantial evidence which can come dangerously close to post hoc ergo propter hoc. In other words, it's not enough to simply observe something being done in the Arab world and then, after the Crusades, seeing it being done in Europe and to therefore conclude that the Crusades were responsible.

To give one very specific, and therefore less controversial example, would be the building of castles, where the crusades seem to have had a direct effect on their construction. As I understand it the authoritative book on this, though I haven't read it, remains T.E. Lawrence's Crusader Castles which was based on his undergrad dissertation The Influence of the Crusades on European Military architecture to the end of the Twelfth Century and was in turn based on extensive fieldwork in both Syria and France.

Another very specific architectural example would be the use of the Squinch which seems to fit the timeline and is a plausible example of such a transfer.

Two further such problems arise, which is that a crusade, itself, is not a particularly good way to promote cultural exchange, considering that it necessarily involves considerable violence and rather ideologically intense combat against your enemies. As such, even where we can look at proposed examples of knowledge exchange from the Middle East to Europe, in the correct time period it's rather difficult to attribute this to the Crusades per se.

So to give one ultra-controversial example of a proposed exchange, George Makdisi posits that the model for European universities were the Madrassas of Islamdom, and that this transfer took place during the crusading era. But his posited transmitters were pilgrim scholars, not crusading warriors.

Secondly, it's worth noting that Spain seems to have been a much more important route for knowledge exchange than cross-Mediterranean or trans-Byzantine exchanges in the Crusading era. It's through Spain, for instance, that paper comes to Europe, and that most translations of ancient Greek texts arrive, not to mention the works of Arabic philosophers themselves. Just as Syriac-speaking Christians were critical in the "translation movement" in Islamdom, so Spanish Jews seem to have been critical for translating these Arabic texts into Latin. This goes back to the nature of the crusades, which is that unless you have someone who actually understands the texts in question it's unlikely that they will value them. That's why engineering developments can seem more plausible than cultural ones.

I'm not an expert on this, but as far as I know the biggest knowledge exchange made possible by the Crusades was the rediscovery of Aristotle in the west. This lead directly to the small renaissance of the 13th century and indirectly to the scientific revolution through the influx of new ideas caused by Aristotelian in the west.

This article tells it in an easy-to-read story: http://www.crisismagazine.com/2013/the-real-significance-of-the-crusades (the source might seem a bit less reliable, but as far as I can tell they place everything in a correct context and everything fits with what I have been taught in the courses I have taken on medieval philosophy)