I know a local who did actually go out looking for students to throw off Prebends Bridge etc.
He did that ~20 years ago, annd it's all changed very much (The Wear is also no longer black from collapsed under-river coal mines).
Ignore stupid mis-informed comments and go into The Fighting Cocks for a drink (or even 2 for £6 brilliant pub lunch), student nights are Wednesday and Monday.
I wouldn't go to that pub (or any Durham club) on Friday/Saturday nights, it's too busy with drunk locals and students will stick to college ents at the weekend anyway. North Road to the Market Square can be a place to avoid Friday/Saturday night, but I do walk through there when I'm going to someone's house. I wouldn't plan to hang about, but it nothing too bad should happen, normally a police van is present on Framwell Gate Bridge. Loveshack area can also be bad.
I really didn't like the look of Gilesgate(out of town, so cheap studeent houses), but it's a look and much safer than other cities.