When I try to resume a file, I am able to download the entire file perfectly. But, when I try to burn the file on a disc or transfer it out of my computer, I get "Data error, cyclic redundancy error" I'm not sure if its a computer problem.

This sounds like an error with your hard drive. You should definitely scan for bad blocks.

how would I fix that, defragment my hard drive?

Scandisk in "Thorough" mode.

For Win 9x|me as Online says, scandisk.

For WIn NT it's chkdsk

Open 'My Computer' > RightClick on your C: drive > Select Properties > Tools > "Error checking" ("Check now")
Check both boxes in the dialog which pops up and click 'start'
Click "Yes" in the popup about open files etc
Reboot.

Or

Start > Run > type cmd.exe{ENTER} > In the command prompt window, type chkdsk C: /F /R{ENTER}
when you see

• Chkdsk cannot run because the volume is in use by another
  process. Would you like to schedule this volume to be
  checked the next time the system restarts? (Y/N)

type Y{ENTER}
Reboot

Do the same for any other hard drives you may have (they probably won't require a reboot).

After running chkdsk on all drives - Defragmant them.

Here (98) CHKDSK is a part of the DOS era and can't even scan FAT32 volumes.

Yeah, but NT has a 32bit version, it still runs in console mode though.
There is no scandisk on NT.