first, nothing in the instructions tell you to set any of the files to 777.
it says to make sure that you have write access to your phpbb files ( 755 )
anyway, now that you have changed them, you need to change them back to what they are supposed to be for security reasons.
in phpbb ( including easymod ) , all folders should be set to 755 and ALL files should be set to 644.
the only exceptions to this are:
1. the avatars folder should be set to 777 if you allow avatar uploads.
2. if you are using the cache, the cache folder should be set to 777
personally, I have never had to change any file permissions when installing EM ( several hundred installations on many different hosts, etc. )