I've tried to download the game three times now, and every time the installer says that the bin file is damaged, and I can't install.

Which game and which bin file?

Neverwinter nights Diamond.
File name is: setup_nwn_diamond_2.1.0.21-1.bin
Every seen anything like it?

Just the clear the obvious, you are running the .exe to do the install.

I have installed from these files and here are the checksums:

File: setup_nwn_diamond_2.1.0.21.exe
CRC-32: f90d7a08
MD4: be999277ef9a5c8e209602efb729ea1f
MD5: cd809b9d22022adb01b0d1d70c5afa8e
SHA-1: d7adc8380bc7e53d87056ef295067a97f0388f80

File: setup_nwn_diamond_2.1.0.21-1.bin
CRC-32: 0f531d3b
MD4: 8e4d155418fa2da4fcaff053700a53db
MD5: ce60bf104cc6082fe79d6f0bd7b48f51
SHA-1: 42764e6ad6584145475f855ff23eb6b298ffea54

Incomplete or corrupted download is the most likely issue. PeterScott's hashes match what I have. Also, why not?

SHA224 is bit overkill just to check file integrity. I only included 4 different ones because I am not sure what people typically use and the tool I added to Windows spits out all 4. You really only need to check 1 of them for a simple file check.

It's too bad GOG doesn't include a checksum, could save people the trouble of re-downloading if they suspect a bad file.

I tried using GOG's downloaded last night, and got the the same result, but it says that I have a checksum error on chunk 152. "Please contact support." Is a checksum error on my end or theirs? And thanks so much for all your help guys, really appreciate it. I love this game :)

Given that you seem to be the only one with this issue, I would say the problem is on your end.

Do a checksum on your files as we suggested.

There is one built into windows, but it is command line:

E:\Downloads\NWN_stuff>CertUtil -hashfile setup_nwn_diamond_2.1.0.21-1.bin MD5
MD5 hash of file setup_nwn_diamond_2.1.0.21-1.bin:
ce 60 bf 10 4c c6 08 2f e7 9d 6f 0b d7 b4 8f 51
CertUtil: -hashfile command completed successfully.

Or download GUI util to do it:
https://checksumgui.codeplex.com

Sure, but who uses CRC-32 or MD4? I could see posting both MD5 & SHA-1. Of course MD5 is broken and SHA-1 well on it's way. Not that you need the hash to be cryptographically secure for checking a download. SHA256 would of course be more common. I did a fresh download last night via Firefox so likely your end.

Like I said, I just cut and pasted the complete output.

MD5 and SHA-1 are both fine in this case, and even CRC-32 or MD4 as well.

This isn't a cryptographic security issue, but just checking for file integrity/corruption issue. Any checksum will do.

It doesn't matter, as long as the .exe and the .bin files are both in the same folder when you run the exe.
Even your desktop would work, though not very practical if to want to keep the installers.

Edit: I thought you meant where do you download to, but in case you mean where to you download from.
setup_nwn_diamond_2.1.0.21-1.bin is half of the classic installer. (that is it doesn't include the Galaxy client.)
Go to your account at the top of the page and then games, and on Neverwinter Nights Diamond click the small down arrow that appears when you move your mouse over the game, then view downloads.
Mine goes straight to the classic installers, but you might have to choose classic installers if this is your first time.
There are two files there, setup_nwn_diamond_2.1.0.21-1.exe and setup_nwn_diamond_2.1.0.21-1.bin