Before the Land Bridge Theory, how was human presence in the Americas explained?

I saw a documentary a few years ago - sorry, no idea who made it - postulating that during the last ice age very early humans could have sailed/walked/paddled along the edge of the ice sheet to reach America from Northern Europe. Why not?

Also, people in boats could be blown over the Atlantic in a storm, supposing they had something to eat in the boat or fishing tackle.

There is a pop science book, The Seven Daughters of Eve, following the spread of humans by tracing mitochondrial DNA, that touches on this theory too.

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Here - I couldn't find the documentary but here is a link

If you google Solutrean_hypothesis you can find some documentaries, but I am at work behind a firewall.

The most sophisticated alternative theory is perhaps that of Florentino Ameghino who postulated an autochthonous evolution of species based on skeletal anthropology and diggings according to the following chart:

German anthropologists, led by Hermann Burmeister, opposed this theory. Burmeister proposed a catastrophe theory of human evolution, similar to that of Cuvier. According to these ideas people evolved not gradually, but very rapidly and independently. William Henry Holmes was active in disputing that there was any evidence humans were in the Americas before the last glaciation, and therefore must have come from elsewhere. Franz Boas disputed this and argued that man came in the interglacial period. Likewise, Daniel Garrison Brinton argued that American man must have arrived in the glacial period and since Siberia was thought to be uninhabited at that time, they could not have a Siberian origin, but must have been Polynesian or some other Asiatic emigration. Brinton also argued that Eskimoes are completely distinct from Siberians, yet occupy the entire Arctic circle, so they and other Americans, who are akin to them, must have had an independent, non-Siberian origin. Also, he disputed that the idea that America was cutoff because Eskimoes circumnavigate with no need for a land bridge.

Prior to these theories, there was a large school of thought that considered Americans to be autochthonous. These theorists included: Levegue, Humboldt, McCullogh, Morton and Quatref*ges. Most of these ideas proposed only semi-autochthony and had some catastrophic or evolutionary component. A few theorists, such as Kaimes, Morton, Nott and Glidden supported fully autochthonous origins with no old world component. Note that many old theories included over-land travel, so the idea that a "land bridge" is a modern idea is somewhat incorrect. The theme that native Americans derived from Siberians of some kind or Mongolians was prevalent very early on.

Just as a side note, the "theories" that Indians derived from Israelites or Phoenicians or some such, were just fantasies by "historians" writing for a public audience, and were not considered serious science by real anthropologists. I have listed above the main scientific thinkers from the pre-modern period who were considered serious scientists.

Early post-contact beliefs contained an unhealthy dose of myths and legends, e.g. Atlantis or that Native Americans descended from the lost tribes of Israel. These were displaced as rationalism developed, but suspicion that the Old World populated the Americas grew over time (for the alternate view, that the Natives had always been in the New World, see @twosheds' excellent answer).

By the 18th and 19th century, various theories were being proposed about various ancient groups sailing to the Americas during antiquity, including:

• Ancient Phoenicians: part of a popular general Semitic colonisation of Americas belief
• Ancient West Africans: some linguists suggested that they created the Olmec civilisation, on the basis of a similarity in writing scripts
• Ancient Chinese: some historians thought Fusang, a mythical land to which expeditions had been sent (notably by the First Emperor), was the Americas.
• According to Mormonism: the Mulekites, the Jaredites, the Nephites, and the Lamanites; essentially an updated version of the lost tribes theory.

These so-called "theories" were pretty half-baked, and the land bridge theory proved to be much more compelling (though its now believed settlement began earlier than that theory thought). But even today some still have currency as fringe theories.

Many prominent men of science in the 19th century believed that the Indians' ancestors had always been in America. This belief draws on the theory of polygenism--that the several races had independent origins as separate species. "Scientific" polygenism also had a religious aspect called "Pre-Adamism." Polygenists/Pre-Adamists didn't need to posit ancient transoceanic travel or land bridges because they just assumed that God had placed Indians in America during Creation.

Some examples: Samuel George Morton studied the Mound Builders and concluded that they were an ancient race, autochthonous to America. Charles Caldwell thought there were four races (Caucasian, Mongolian, American Indian, and African), each a different species, created separately by God, while Charles Pickering thought there were eleven separate races. Also see Louis Agassiz and Josiah Clark Nott. Note that many American Indians at the time favored theories of polygenesis, as these were more consistent with their own origin stories.

Semaphore's answer ably covers theories based on polygenism's rival, monogenism--that is, the assumption that all humans have a single origin--Adam and Eve, Africa, etc. Assuming monogenism, you then need to explain how American Indians got to the New World from the Old World. But if you believe in polygenism, no additional explanation is necessary.