basename with spaces in a bash script?

The problem is that $1 in

MYBASENAME="`basename $1`" 

is not quoted. Use this instead:

MYBASENAME="$(basename "$1")"

You're missing one set of quotes!

MYBASENAME="`basename \"$1\"`"

That'll fix your problem.

In the case where the assignment is a single command substitution you do not need to quote the command substitution. The shell does not perform word splitting for variable assignments.

MYBASENAME=$(basename "$1")

is all it takes. You should get into the habit of using $() instead of backticks because $() nests more easily (it's POSIX, btw., and all modern shells support it.)

PS: You should try to not write bash scripts. Try writing shell scripts. The difference being the absence of bashisms, zshisms, etc. Just like for C, portability is a desired feature of scripts, especially if it can be attained easily. Your script does not use any bashisms, so I'd write #!/bin/sh instead. For the nit pickers: Yes, I know, old SunOS and Solaris /bin/sh do not understand $() but the /usr/xpg4/bin/sh is a POSIX shell.