Spaced out enemies
Once you climb down the stairs below the boat, and into the first level of the caverns, you will notice that you have to jump from one platform to another in order to get through. However, most of the platforms are occupied by enemies. This can be a problem for many characters, but not for Simon Belmont. His whip can easily extend the distance to the next platform and clear the way for the next leap. Use him to create a safe path from one platform to the next. If you don't have Simon yet, you may wish to consider going back to Castle Dracula to retrieve him, or using one of your character's special weapons to shoot across the gap. While you attempt to reach the other side, you will undoubtedly encounter one of the Jellyfish. If one happens to cross your path and can be hit, do your best to defeat it. Otherwise, ignore it completely and watch out for the bubbles it fires at you.
Plotting a course
If you take a look at the map, you will notice that the path to the lowest level branches quite a bit. It is a good idea to plan your route through the stage in order to avoid back-tracking as much as possible. From the upper portion of the cavern (where you can collect Konami Man's Beam Gun), you can go left or right. The left path splits up again after you climb down. The order of priorities is collecting the key, rescuing Mikey, and collecting the slingshot, Mikey's special weapon. The key is obtainable by taking the left path, and continuing left along that path. You will be forced to backtrack after you collect the key (unless for some reason you have already collected the Mantle and can fly over the lava). After getting the key, return to the upper portion and follow the right path. This will take you to an area where you can rescue Mikey. Unfortunately, once you have him, you'll have to return to the upper portion again, take the left path down, and then follow the staircases straight down in order to reach Mikey's slingshot.
With all of the criss-crossing that you have to do in order to complete this stage, you will no doubt encounter the stage's enemies multiple times. The fire-breathing dinosaurs are probably the most dangerous to contend with. Avoid them entirely is a safe strategy, but sometimes you will be forced to deal with them. Try to wait for a break in their fire breathing, and then run up to attack them. Hit them frequently enough to freeze them in place and defeat them before they get the opportunity to breath on you again.
Use the warp
After all the running around that you do in this stage, it is a big time saver to use the warp found in this stage to get you out of Goon Docks and return you to Dr. Cinnamon's lab. The warp can be found directly below the screen where you rescued Mikey. It is easy to reach if, after collecting Mikey's slingshot, you walk to the far right of the bottom of the cavern, and up two screens. Then you will find yourself back near the place where you rescued Mikey. Walk all the way to the left and kneel down in the corner for a quick trip home.