Hi. I've been in Bratislava for about 10 days now. So far I've felt reasonably safe during the day. The only time it got a bit weird was when I was passing through a group of drunk homeless men and they started shouting at me something in Slovak and I just kept walking and then they booed me. Even then though I didn't feel unsafe because it was morning, there were many people around so I knew nothing bad would happen.
During the night though I have only been in the city centre and it was pretty cool, night life seemed to be booming and no incidents happened or have been observed by me.
As for criminality and social problems, you can find all of the statistics online as well as locals talking about social problems. I personally didn't look for criminality statistics but I did check the racism levels and they are interesting, I guess. Most racism that exists is towards Roma (gypsies) and other people with brown skin. Which is sad but at the same time not too surprising as the situation is quite similar in most of the eastern countries, from north to south. I think it's mainly because they haven't really encountered many people from other races and tend to believe the worst stereotypes out there. I am sure this will change but for now it is how it is.
Having said that, it is what I've read. Not looking different from them myself, I haven't encountered any racism (possibly because they don't know I'm an expat but I really doubt it) and the people that I've met so far seem to be open minded, they love travelling so they're not new to internationality of it all and haven't shown any racist tendencies towards me or anyone else.
Hope this helps and doesn't scare you off, the city is pretty wonderful and a must see!