How do I not screw up the best ending in Cave Story?

I will say up front that I don't think that Easy difficulty affects this, but I've only played the original. However, unless they felt like being callous, it shouldn't have an impact (considering you can access the Best Ending in other alternate game modes as well).

The Best Ending isn't incredibly difficult unless you (like me) are playing at 3 Health. If you've been picking up all the health increases and arm yourself properly, you should be able to accomplish it eventually. So I would recommend going for it because it is a nice ending. As well, even if you meet the requirements for the best ending, you can always opt to still take the normal ending (or even the bad ending).

You basically need two things to get the Best Ending. This is obviously a spoiler but you're literally asking for it, and a 90% spoiler post looks stupid. You need the Booster 2.0 and you need Curly. All the big points you need to make sure to not screw up are all in bold below.

Booster 2.0

• For Professor Booster to survive, he needs to keep the Booster 0.8. So you can't take it from him - this means when you see him fall past you in the labyrinth, don't drop down. Instead, jump across the gap (you can do it with a good moving start or just use Machine Gun lv. 3 if you have it) and continue onwards.
• Later on when you return to the Mimiga Village, Professor Booster will warp into Arthur's house. He will have completed the Booster 2.0 and will give it to you. Objective complete.

Curly

• When you access the Labyrinth Core room for the first time, do not immediately investigate the robots (which triggers the boss fight). Instead, drop down to the bottom right corner of the room, there should be something shiny. Pick up that Tow Rope, you'll need it, and it washes away if you don't pick it up now.
• When you defeat the Labyrinth Core boss fight, Misery floods the room and Curly gives you her oxygen tank. If you picked up the Tow Rope I had just told you to, go examine Curly and you should use the item to grab Curly and bring her with you.
• In the ensuing water ways sequence, make sure to stop at the cabin on the way. Put Curly on the bed and examine the bookshelf to find instructions on how to drain the water out of her. Go do just that, and remember to take her with you when you leave. She'll drift away after fighting Ironhead, this is expected.
• Much later on in the Plantation area, head to the bottom right of the whole map and you'll find Curly and another guy. Talk to both of them to get caught up on the story and learn about a mushroom you need to grab in the Mimiga Graveyard.
• Backtrack to the Mimiga Graveyard and use your fancy new Booster 2.0 to get to the previously inaccessible door. There, you'll find a mushroom. Pass his quizzical bombardment, and you'll get a Mushroom Badge. Examine that badge and then talk to that conniving mushroom so that you can get the real mushroom. By force.
• Return to the Plantation and give Curly the mushroom. The ensuing conversation gives you a ton of backstory and also completes the requirements to have Curly. After you finish your conversation, talk to her again to get the Iron Bond.

With both Curly in tow and the Booster 2.0, you gain access to the Best Ending. This does two things you will want to be aware of, though.

First, it does indeed ramp up the difficulty of the Last Cave section. It ends up filled with more spikes, and all of your weapon experience levels are dropped to 1 at the start. It isn't a severe ramp in difficulty, though, just a taste of what is to come. This shouldn't worry you as you'll have the Booster 2.0, which is worth it, and the difficulty spike isn't that strong.

Second, after you beat the final boss, during your escape, you should pass by the house you had last used as a save point. This door would be closed if you failed the requirements. Inside will be another door that leads to the Sacred Grounds. This is an extra dungeon that you must clear, upon which you'll receive the Best Ending. You do get the option to use the Save Point again so you don't have to re-fight the final boss. As I mentioned earlier, though, this is optional and so if you want to you can always ignore the house and settle for the Normal Ending.

A tip for jumping over that cliff where professor Booster falls down, without Machine Gun:

• Use that savepoint first, because if you fail this jump, you won't get back up without booster 0.8 which means you fail getting 2.0 which means you fail at Best ending.
• At the left side (where you're coming from) there is a block with a red dot. Run towards the cliff to get momentum and jump right BEFORE (or right over) that red dot, press jump button all the way through the jump and you should clear the jump fine.

The dot is there in Cave Story+. I can't remember if it was there in original CS.

And another thing, if you go back to that prefab building to enter the Sacred Grounds, you can save and still get to normal ending since checking the bookshelf will give you choice to rewind a bit. So if the doorway to that house is open, there's really no reason not to go in.