They were on the whole pretty unconcious of the threat. The only prior knowledge Europe had had of the Mongols came from garbled stories that Crusaders had heard about Mongol conquests in the East. As they had very little idea as to what was actually going on they came up with a distinctly distorted image. The rumors of Christians being present in these mysterious armies (indeed many Mongols were Nestorians, an eastern branch of Christianity) and the fact that these armies were also attacking the Islamic world lead them to see them as the armies of Prester John. Prester John was a European tale of a rich and powerful Christian king who ruled far off in the East.

So clearly when the Mongols rock up in Europe the Europeans have no idea what's going on. This is reflected in the panicked chronicles written at the time, the gist of them is that they've been hit by suprise and don't know what's going on. This is reflected in the monk's attempts to work out who these invaders are. The sheer foreigness and lack of knowledge of the Mongols meant we get some interesting guesses. We have suggestions ranging from their being the armies of the Antichrist heralding the apocalypse to their being one of the lost tribes of Israel (this idea inspires a few pogroms). This complete lack of knowledge indicates that Europe had no idea what hit it.

When I get home I'll see if I can find more details for you.

Well, Arghun Khan of the Ilkhanate tried to make allies with France and the Pope against Islam in 1190. He viewed Christianity and the native Mongol religion of Tengrism as compatible and joined in purpose in defeating Muslim expansion. The Pope demanded that the Ilkhanate convert to Christianity, which Arghun Khan either didn't understand or outright refused. The alliance ended up not going through, but the written correspondence on the issue of this Alliance is one of the best primary documents we have describing contemporary Mongol religion.