At the risk of sounding somewhat non-committal, a bit of both. While I can't speak to purely military history, the Mongols had a complicated diplomatic relationship with the crusaders.

Up until 1260, many crusaders feared the Mongols even more than they did the Mamluks—which is why the barons of Acre agreed to what was essentially an armistice with their Muslim rivals, allowing the Mamluk armies to cross their territory on their way to the Battle of ʿAyn Jalut. (This despite a Mongol embassy to Cyprus in 1248-49, as Western Europeans still remembered the Mongols' depredations in Hungary and Poland in the earlier 1240s.) The Mongols' defeat at ʿAyn Jalut caused the Ilkhans—the viceregal dynasty in charge of Iran and the Middle East (which eventually broke off to do its own thing)—to see the crusaders as potential allies in the fight against the Mamluks.

Thus began a period of tentative diplomacy between the Ilkhans and the crusaders, who explored the idea for a joint attack on the Mamluk holdings—the Christians from the west, and the Mongols from the east.^(1) But these plans never really came to fruition, for a number of reasons. (Internal strife both among the kingdoms of Western Europe and within the Mongol Empire and its successors; logistical difficulties; Western distrust of the Mongol world-imperial project.) Obviously, the crusades ended in failure despite the hopes of the Ilkhans and certain Western actors, and enthusiasm died out in the West.

That said, there were certainly periods in which Mongol intervention "helped" the crusades, if only temporarily. Peter Jackson notes that Ghazan Khan's occupation of Syria in 1299-1300 represented the "high-water mark of Mongol-Latin relations," and the Mongol conquest of Jerusalem was celebrated at many Christian courts. But this was a rather ephemeral triumph; by mid-1300 the Mongols had pulled out of the area.

So strategically, did the Mongols help or hurt the crusaders? Probably a slight help, insofar as they drew Mamluk pressure to the east, away from the few remaining crusader states, and prolonged the crusading footholds in the Levant for a few years. But more importantly, they had a great moral effect for pro-crusading parties back in Europe.

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^(1) This period lasted until the early 14th century; the last known embassy to a Western monarch was in 1313, and the Ilkhan Abu Saʾid finally made peace with the Mamluks in 1320.

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SOURCES / FURTHER READING

There's a lot of literature on Mongol-crusader diplomacy, so I'll stick to mentioning Peter Jackson's The Mongols and the West, 1221-1410, which is a pretty comprehensive overview of the subject. It has a chapter summing up much of the previous work on Mongol-crusader relations: "An Ally Against Islam: The Mongols in the Near East" (165-195).

See also (it's a bit older than Jackson's work, but more focused on your question) Denis Sinor's chapter on "The Mongols and Western Europe" in History of the Crusades, ed. Kenneth M. Setton, vol. III (513-544); and Sinor's article "The Mongols in the West" (Journal of Asian History 33[1], 1999: 1-44).

Also of interest (again, more from the perspective of diplomatic history) is Denise Aigle's article "The Letters of Eljigidei, Hülegü and Abaqa: Mongol overtures or Christian Ventriloquism?" (Inner Asia 7[2], 2005: 143-162).