I've done some experimentation...
There seem to be maps that are generally 'bad' to save in. For example, saving in map 26 (this is the small room in grasstown that holds a save point, the one closest to the power room) will screw up your save, and the game will refuse to load a savefile that contains map 26 even after using the Save Fixer on it.
Other maps are good places to save. Saving in Arthur's house, which is actually map 1, will still corrupt your save but this time it should be easily recoverable by the Save Fixer. Santa's House and Chaco's House are also good places to save. There are many other good maps, but I don't have a list.
For all of these maps, the Save Fixer can generally recover your file successfully.
Unfortunately, it looks like the prefab house (either before or after final bosses, or both), is a bad place to save.
First cave might also be a bad place to save, but that doesn't really matter since you can get to Mimiga village easily.
The way to get through Cave Story+ is to be picky about which maps you save in. After I defeated Balrog in the power room, I did NOT save in the closest save room because that would permanently ruin my save. Instead, I went back to Chaco's House and saved there. The save was still corrupted, but lo and behold, it was still fixable, unlike the one for map 26.