How to erase photos from my iPhone 5S?
Plug device into Mac. Open Image Capture (in the Applications folder), select device in sidebar. Check the option at the bottom that says "delete on import" then import all files to the selected folder or program.

I used to do this manually, only deleting older photos, but with the iCloud Photo Stream, you will always have the last 1000 pictures available, so there's no reason not to just delete all after import.

You may end up with duplicate photos. There are several tools you can use to resolve dupes. Personally, I dump all imported photos into a specific directory, then run a dupe resolver, then add the new images into my photo management programs. (Not an iPhoto user because it doesn't sync well between Macs. Don't know if it has any in-built dupe resolution. YMMV.)