This can not be done in the Management Studio GUI, but must be done by T-SQL as the GUI fails on a UNC reference to the backup file.
The UNC reference is on a share I create on the backup disk. I prefer to name the share "SQLBackup".
I have looked around for Best Practices on file shares and security. The best description I found is by Derek Melber (see Reference), but I have some issues...
It makes sense in general to control the details in access and rights i NTFS, and make a general access on the share. Derek suggests to grant access on the share to Authenticated Users in the domain and give Full Control rights. This should be done while you keep in mind that a Windows default right is that Everyone can Read.
When we are dealing with database backup files, that contain sensible data, I find this access a little too "broad" as is grant access to too many accounts.
Instead I have given specific Read access to the share, and depends on the general Read access to Everyone.
If Everyone is removed, and this I have seen in several organisations, you have to take care of the file access and rights also.
And again: Keep in mind that the backup files are to be handled as sensitive data.