The Royal Navy was more powerful than the American navy at the time the doctrine was declared, so what stopped the British or some other power from simply ignoring it and establishing anyways? Surely the US couldn't defend the entire coast of Latin America from European landings?
You're correct, the USA in the early half of the nineteenth century did not really have the military strength to enforce the Monroe Doctrine, however, in practice, it was actually enforced by the Royal Navy, in so far as it was a military policy, though it was a diplomatic one as well.
What came to be known as the 'Monroe Doctrine' (though it wasn't called that initially) was a response to the collapse of the Spanish Empire in South America and the decolonisation of it's former colonies. The British, for whom Spain had been a key rival, had arguably as great or greater interest in enforcing the doctrine than the USA.
While the USA did not issue a joint statement on the issue with the British, despite the wishes of British politicians, both parties were interested in enforcing it, and in practice it was enforced by then both.
There's some more information about it here: https://history.state.gov/milestones/1801-1829/monroe