Post it on a personal website where you have control of the files (such as Google Drive, or OneDrive). That way once the dump file reaches it's intended target, it can then be deleted. You could even password protect it on such a site (and send a private message to the target with the password).
For the most part, the people who actually read the dump file could care less about looking in your dumps for passwords or personal info. I've personally seen 1000's of dump files here and never even thought of doing such a thing, but that's not saying that some other person could stumble upon a thread and try to get such info.
Another option is to have some type of binary editor that allows you to search and replace the text (without screwing the file up). With this idea, you could search for some of the passwords you think might have been in memory at the time, and replace the password with another word (of the same length). This can't be done in your average text editor, it will not re-save the file correctly. You would need a binary editor.
Minidumps don't contain that much memory. They contain the stack info, driver info, BIOS info and are usually only a few megabytes long.
A full dump, like said above, might contain everything in memory at the time of the crash.
I personally prefer minidumps over the full memory.dmp files but that's because I'm no expert at reading dump files and many times the full memory dumps are corrupted.