when the US declared that ‘everything on this side of the Atlantic was ours’, how could they enforce it? Our navy was small compared to French, English and even Spanish Navies, especially if 2 or 3 all came over at same time.
Yes. It was only enforceable as long as preventing other European nations from continuing their American expansion benefited Great Britain… and GB was glad for a rationale to prevent this expansion. As long as the British Navy supported the concept, the Monroe Doctrine held.
I looked up the Wikipedia article to remember when exactly it took effect, as I thought maybe the navies were preoccupied with other stuff at the time. That doesn’t appear to be the case, but there are two great paragraphs that address this question:
Because the U.S. lacked both a credible navy and army at the time, the doctrine was largely disregarded internationally. Prince Metternich of Austria was angered by the statement, and wrote privately that the doctrine was a “new act of revolt” by the U.S. that would grant “new strength to the apostles of sedition and reanimate the courage of every conspirator.”
The doctrine, however, met with tacit British approval. They enforced it tactically as part of the wider Pax Britannica, which included enforcement of the neutrality of the seas. This was in line with the developing British policy of laissez-faire free trade against mercantilism. Fast-growing British industry sought markets for its manufactured goods, and, if the newly independent Latin American states became Spanish colonies again, British access to these markets would be cut off by Spanish mercantilist policy.
My understanding is that the British provided the muscle behind the Monroe Doctrine, as they were keen on opening up Latin American markets for trade and influence. The original Monroe Doctrine was a statement of opposition to the restoration of colonial empires in Latin America, mainly directed at Spain and Portugal, and later, France. It was only later in the nineteenth century that the United States assumed a more hegemonic role in Latin America. The “hemisphere is ours” interpretation of the Monroe Doctrine really emerged with the Spanish American War and Roosevelt Corollary, which I think is what the OP identifies as the main thrust of the “Monroe Doctrine.”
Even then, the British probably retained more sway over more distant countries such as Chile and Peru through World War I and perhaps World War II.