In short, not overly in the long term. Those of Eastern Europe were terrified of the Mongols but only after they had met them. The first plausible time frame for Western Europe to know of the Mongols (or at least the HRE and west) is after Subotai's cavalry raid in 1221. This passed through the Caucasus, home to Georgia, a Christian nation obligated to provide troops for an upcoming crusade, who promptly met the Mongols twice in combat and were slaughtered. This led to a letter to the Pope from the new Queen, Rusadan, which I'll partly reproduce from Richard. A. Gabriel's book^1:

a savage people of Tartar, hellish of aspect, as voracious as wolves in their hunger for spoils as brave as lions, have invaded my country. . . . The brave knighthood of Georgia has hunted them down out of the country, killing 25,000 of the invaders. But alas, we are no longer in a position to take up the Cross as we had promised your Holiness to do

The Mongols then met the Venetians on the Black Sea, whom with they traded maps in return for the Mongols destroying non Venetian trading posts in lands they visited. The interactions with the West after this waited till 1238. An embassy from the Saracens^2 arrived in Europe to solicit aid from the Europeans against the Mongols. Unsurprisingly, it was shown little welcome.

After this, further contact was prompted by the invasion of Eastern Europe in 1241 following the invasion of the Russian region the previous year. The defeat of both Poland and Hungary at least awoke Western Europe to the Mongols, who got as far as scouting towards Vienna before they were recalled due to Ogedei's death. As noted by the Grand Master of the Templars in a letter to the King of France, Louis IX, no army of any significance stood between the Mongols and France post Leignitz and the Battle of the Sajo river. So at this point the danger to Western Europe was great, although how great depends on how well you think the Mongols would have coped with the dense forests and castle's of Germany.

As a result of that invasion and subsequent recall, the Pope sent out envoys as a result after the council of Lyon in 1245, which allowed the West to learn of Mongol military strength due to John of Plank Carpini and others. But the Mongols still viewed European nations as potential member of the Mongol world empire as shown by Guyuk's letter to the Pope in the late 1240's. However this hostility dropped off following the nominal break up of the Empire near European influnce into the Golden Horde in Russia and the Ilkhanate in Syria near Crusader states. Hulegu of the Ilkhanate even initiated alliance offers against the Mamluks to these Crusader states and such offers were repeated over time. The Golden Horde meanwhile continued to attack Poland and Hungary, taking loot but not major land gains, as Hungary fortified following the initial Mongol withdrawal. It's fair to say that after 1259 (Monkhe's death) and the splitting up of the Empire the Golden Horde alone could not threaten Europe to such a degree that Western Europe would be involved.

So in summary, Western Europe in 1242 was not united enough to face the Mongols as they made their way West out of Hungary and Poland, and may well have been ravaged by the horde. The Pope^3 was locked in disputes with the HRE, and no one would support crusades against the threat. However, after this initial invasion no such threat ever appeared so far west again. Richard. A. Gabriel wrote that:

the great trauma of the Mongol invasion that frightened the princes and churchmen of Europe to death was quickly forgotten, with ... no discernable affect on western military practice.

So the in essence threat was only large once, and the West quickly forgot it in amongst the petty feudal squabbles in France and Germany.

Apologies if this wasn't exactly what you were looking for, it may be a little around the topic in parts!

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^1: Richard. A. Gabriel - Subotai the Valiant

^2: Matthew Paris - Chronica Majora (taken from The Mongols by David Morgan)

^3: John Man - The Mongol Empire