How can I check if a compiled binary is 32-bit or 64-bit?

Check out the file command.

$ file /usr/bin/grep
/usr/bin/grep: Mach-O universal binary with 2 architectures
/usr/bin/grep (for architecture x86_64):    Mach-O 64-bit executable x86_64
/usr/bin/grep (for architecture i386):  Mach-O executable i386

The lipo binary also does this.

lipo -archs

$ lipo -archs 'Adobe Photoshop Lightroom 5'
x86_64 

lipo -info

$ lipo -info 'Adobe Photoshop Lightroom 5'
Non-fat file: Adobe Photoshop Lightroom 5 is architecture: x86_64