This one is interesting as someone asked me this question as to what permission does the Style Resource Readers group need access to?
Here is what you need to know about style resource readers group.
When you create a new site collection SharePoint assigns the required permission to Style Resource Readers group. By default it will read Limited Access. You do not need to modify anything with Style Resource Reader group unless you are planning on adding users to that group. Personally I will advise against it.
Limited Access in SharePoint means that there are permission assigned to a user or group on one or more site artifacts i.e. list, folder, item but no permission is assigned to access site.
So Style Resource Readers group out of the box has the Read permission to master page gallery. If you read the description it reads.
“Members of this group are given read permission to the master page gallery and the Restricted read permission to the Style Library. By default, all authenticated users are a member of this group.”
So don’t worry about this group and leave it as is given SharePoint already assigned required permission at the master page gallery and style library level for this group.