Minecraft wiki entry for Defeating_a_monument, associated text: Go in the front door. You might throw an ender pearl to get there quickly, dive from above, or run along the seafloor with Depth Strider III-enchanted boots.
Mine through the roof above the elder guardians' rooms. It helps to drop blocks of sand around the gazebo at the top to build a quick bunker around the site. Clear the water out using slime blocks, dig a hole in the roof and shoot the top elder guardian with a bow to kill it and get your first sponge, and work your way down. In Java Edition, use milk to counteract the Mining Fatigue. In Bedrock Edition, milk removes Mining Fatigue for no more than one second so milk isn't useful; instead, use the correct tool (shovel for sand, pickaxe for stone, hoe for sponges, etc.) for each block you mine (gold tools are most efficient but wear out quickly), preferably enchanted with Efficiency, and you can actually make progress under constant Mining Fatigue.
Tunnel up from below. For this you'll almost certainly need milk to counteract the Mining Fatigue. An efficient way to use this method would be to bring a cow and a few buckets. One of the safest ways to do this is to mine below the ocean's floor until you get to below the structure. From there you can mine up one of the 4x4 columns using a 2x2 spiral staircase, or simply using ladders if you want. Ideally, you bring a cow and a few buckets to a base you make underneath the monument, that way you can get milk much quicker. It takes a lot of milk.
In Java Edition, if the player creates a platform over the apex of the monument, TNT can be activated and dropped down and an elder guardian can be killed without any structural damage. If structural damage is preferred, sand or gravel can be dropped after the activated TNT, which will make the TNT explode inside the sand block, and structural damage will be dealt. This method can be repeated over the left and right wing of the monument, to kill the rest of the elder guardians. This can also be done underwater, by placing a sand/gravel block on top of the placed TNT before detonation.