Not very. According the the [Annual Report from the Secretary of the Navy, 1823] (https://www.history.navy.mil/research/library/online-reading-room/title-list-alphabetically/a/secnav-reports/1823.html), our Navy was spread around the globe already - the Mediterranean, African Coast, Pacific coast of South America, Caribbean, and the American coast. Our ship of the line USS Franklin - one of about a half dozen we had in service - was launched in 1815 with 74 guns, while the British launched a 120 gun ship in 1823, the year the doctrine was issued (our three largest ships of the line were 90 guns each). In fact the British launched six 120 gun ships from 1808-1823 alone, even more in the next decade. The French had launched several 118 gun ships in the same time frame (8 from 1806-1824). Half of our ships, namely frigates, needed some form of repairs and some are even listed as "rotten" or "decayed and sunk". The Royal Navy was considerably stronger than ours and to launch any meaningful effort would have meant rushing production of 5 ships of the line and 5 frigates in varying stages of completion and totally unfitted as well as sacrificing our presence in other theatres by recalling those ships. Additionally our presence in Florida and the Caribbean, where the majority of our forces already were (which were mostly pirate tracking sloops and barges), had seen the death of numerous officers mainly from disease. 46 Navy officers were listed in the report to the President as having died between Jan 1 and Nov of 1823 while in service.

But the bigger question becomes what did it all really mean when Monroe issued it? It was basically just asking Europe not to meddle. The British (and several European countries) had learned true power is in trade and moved towards that goal anyway. The British wanted, and suggested, issuing a joint release by England and America in part due to Russian chatter over helping Spain to recieve reciprocation on their claim of the western portion of North America. We declined and issued the [Monroe Doctrine] (https://www.ourdocuments.gov/doc.php?flash=false&doc=23&page=transcript) solo but acknowledged their mutual desire to the goal in the doctrine itself. With British support it was unlikely any would attempt to conquer the volatile areas that were free to trade with otherwise and we knew this going in. In fact Monroe consulted the other "Virginians", Thomas Jefferson and James Madison, who concurred with the theory any who violated would face British forces yet we would make the claim ourselves and not as their assistant in the endeavour.

None the less, as time went on European powers did influence South America (and the French in North America, which is a whole 'nother story that gave us the American "holiday" Cinco de Mayo) several times to which we did pretty much nothing. By the late 19th century our military might was well enough built up to act as a significant deterrent, but it merely was a deterrent in theory as we never really prevented any European powers from meddling, including the British in the Falklands as recently as the 1980s (technically we abandoned the doctrine under FDRs good neighbor policy).

By the early 1900s Teddy attached his [Corollary] (https://millercenter.org/the-presidency/presidential-speeches/december-6-1904-fourth-annual-message) to the doctrine and then we had justification to intervene directly for the betterment of the inhabitants, and we did do that in several South and Central American countries like Cuba, Nicaragua, and Haiti. His justification;

Chronic wrong-doings, or an impotence which results in a general loosening of the ties of civilized society, may in America, as elsewhere, ultimately require intervention by some civilized nation, and in the Western Hemisphere the adherence of the United States to the Monroe Doctrine may force the United States, however reluctantly, in flagrant cases of such wrongdoing or impotence, to the exercise of an international police power...

Circling back, the French never took it seriously and did what they wanted. When they invaded Mexico in 1862 we couldn't do a whole lot. If we did (which we had no bandwidth to do) we risked France recognizing, or even directly assisting, the CSA. We did smuggle arms to the rebels that ultimately helped push the French from Mexico, but it was certainly no grand army or navy action protecting from European invasion.

Robert Remini wrote (imo) a great book on J.Q. Adams that has a chapter mainly devoted to the doctrine and its original motivations. It also covers the rest of his very interesting life and is titled John Quincy Adams: The American Presidents Series: The 6th President, 1825-1829. Another well recieved book on the man that had more to do with it than Monroe himself did is John Quincy Adams: Militant Spirit by James Taub - unfortunately I have yet to read this one but have had it recommended to me by others.