Stirrups were available to Western Europeans well before (about five centuries before, although exact dates are difficult to determine) the Mongols arrived. It's a little tricky to say what influence the Mongols had on Western Europe, considering that they never really got to Western Europe (other than diplomatic delegations to France and such).

However, some scholars attribute the transmission of gunpowder from China to Europe to the Mongol invasions. What effect did guns have Western European cavalry? In the very long term -- by the early 20th century -- guns drove cavalry from the battlefield. In the medium term, though, you could make the case that guns rescued the relevance of cavalry. This case isn't regarded as established fact, but since you ask about the effect of the Mongols on Western European heavy cavalry, and not their effect on siege warfare or infantry tactics, I'll present it.

As Western European plate armor became increasingly impervious to traditional missiles (arrows, crossbow quarrels), and infantry became more organized and disciplined (and impervious to heavy cavalry), the battlefield increasingly looked like it would become dominated by massed formations of heavy infantry, as with the War of the Roses (where dismounted men-at-arms were the dominant force in most battles). (NB: Gunpowder was used in the War of the Roses, but on a small scale that didn't disrupt tactics much.)

In this narrative, guns overturned the incipient tactical dominance of heavy infantry in two ways. First, infantry arquebusiers were a direct threat to heavy infantrymen. Second, cavalry themselves carrying small guns (i.e., pistols) could shoot them from out of range.

Infantry gunners were also vulnerable to the traditional cavalry charge, with cavalry often able to charge them with lances and sabers given their long reloading times and uncertain accuracy. So, although the age of pike-and-shot was generally somewhat infantry dominated, cavalry retained more of a role than it would have if gunpowder weren't around and it were an era of pike without shot.

However, at this point I'm really talking about the technology of gunpowder, not the Mongols per se. Even if the Mongols hadn't transmitted to Europe (if they were, in fact, the ones who did), it's hard to believe that Europeans would have been permanently ignorant of it. At any rate, it's rather speculative to claim that cavalry would have withered into an American Civil War-style force of scouts/messengers/picketers/mounted infantry without gunpowder.