Helsinki is a very safe city, but you should keep an eye on your possessions and luggage at all times, just like in any other big city nowadays. This is especially true during the summer in crowded trams (3T/3B are the sightseeing trams) and crowded places.

Kaisaniemi Park near the train station should be avoided during nighttime as there have been some disturbances. Also at night the drunks and vagrants hanging around the train station may be a bit intimidating, but just keep a hold of your handbags and mobile phones and you´ll be fine.

 There are plenty of people walking around the center of town during the night, and it is very safe to do so, also for women.  

 Drunks

Unfortunately, there are quite a few drunks in Helsinki. Occasionally on the trams, you will get a natural wiff, and it's possible that a person smelling of alcohol starts to make conversation with you. Rule of thumb: the more down and out the drunk seems, the less likely it is that you have anything to worry about.  These drunks are alcoholics, but usually do not have mental problems as such. 

To deal with drunks: ignore them, or move away. Look away for as long as possible. Do not start a conversation. Be aware that many drunks in Helsinki are relatively fluent in English, and some are able to speak Swedish and German too. If you are a non-English speaker, speak to your friends in that language, and ignore the drunk. 

If you can't keep quiet, be polite and respectful. Usually, you will have no problems, although sometimes the apologies for rude behaviour can be long and annoying.