Mounted men had some advantages over the infantryman. Their horses afforded them mobility, and a high position to attack from, in addition to the psychological impact of the horse. However, the effectiveness of cavalry in the Americas has probably been overrated, and they were far less decisive than is popularly imagined, at least during the Mexican campaign (with which I am most familiar). The first thing to point out is that there were not many of them. Cortes landed with only 16 horses. Even with reinforcements the number of mounted men never even reached 100, and soon fell.

The second problem is that Mexican weaponry was capable of fighting off mounted fighters. Several sources, including Diaz, the Anonymous Conqueror, and Diego Duran, record several instances of warriors (usually Tlaxcallan) killing horses with two-handed swords and long spears. The Mexica also countered cavalry with long pikes tipped with Spanish swords.

Tactically, Mesoamerican's were not as vulnerable you might think. Diaz describes Tlaxcallan warriors as grabbing the riders' lances. Cortes responded by having his horsemen harry the Tlaxcallans, rather than charge home. He also describes warriors as responding to horsemen by closing ranks to prevent them getting into their formations. The Mexica also used field fortifications which they would lure Spanish horsemen into.

Note that this mainly pertains to Mesoamerica. Horses had a different, but not necessarily greater, impact in other parts of the Americas. For example, Inca warriors were much more vulnerable than Mexicans, as they fought with shorter weapons. However, the Andes was even more inhospitable to horses, and roadside ambushes proved quite effective against them.

Historians have tended to attribute a staring role to horsemen as an easy explanation for the Spanish Conquest. Horse, along with guns and steel, are something that can be pointed to as a clear difference between Spanish and Indigenous forces. Thus, it provides a simple and clear explanation for a complex event.

Sources:

Díaz, Bernal, The Conquest of New Spain, tr. J.M. Cohen, (London: Penguin Books, 1963)

Durán, Diego, History of the Indies of New Spain, tr. Doris Heyden, (Norman: University of Oklahoma Press, 1994)

Mann, Charles C. 1491: New Revelations of the Americas before Columbus, 2nd ed., (New York: Random House, Inc, 2011)

Restall, Matthew, Seven Myths of the Spanish Conquest, (New York: Oxford University Press, 2003)

The Anonymous Conqueror, Narrative of Some Things of New Spain and of the Great City of Temestitan, Mexico, (ed.) Christensen, Alec, obtained from: http://www.famsi.org/research/christensen/anon_con/, Accessed 5, March 2012.