ut 是连词,承接上面的句子.
主语:reputaion
谓语:fall
句子可以缩为:reputaion has fallen .
over the past thirty years :时间状语.
with stories of rampant crime making headlines around the world:后置定语
全句译为:但是随着犯罪故事在全球报纸飞快地上了头条,它的那些“好”名声在过去的三十年也完了.
———————————————————————————————————
应邀再把with stories of rampant crime making headlines around the world 再分析一下.
不是像你说的那么分的,应该是这么断:with )(stories of rampant crime making headlines around the world
with 带着的是一个宾语从句,这个句子可以缩为:story makes headline.
看明白了吗?