watch carefully, WHERE your hotel is...
The area around the Hauptbahnhof is certainly NOT a good place to stay for a family with children, who might wonder, what the many drug-addicts or prostitues are doing there all day and night...
East of the Hauptbahnhof only Kirchenallee + Hachmannplatz and SteintorPLATZ are still good adresses to stay in a hotel.
The rest of St Georg, Adenauerallee, Steindamm, Bremer Reihe, Hansaplatz etc. will have many rather cheap hotels and hostels, BUT I may not recommend them for families and single women travelling alone.
For men I dont see any problem, as long as you stay away from the drug-addicts. This area is not really under police-control all day long...
Reeperbahn has some cheap and rather popular hotels, like Hotel Stern, it is in the very centre of the "sinful area", but at least under police-control, so you will not be in danger and you are really close to many great theatres !
...there are so many elegant hotels in many areas - but if you choose the wrong area in Hamburg it might be a problem. - Simply try to avoid to have a hotel at Steindamm and all the areas around it
Hamburg is a typical European city and more or less the same concerning safety standars. The former SocialDemocratic/Green government was quite relaxed on open drug dealing which angered many citizens and led to its dismissal. The red light district in St. Pauli has become less dangerous in the recent two decades, but there still is a drug scene in some of the districts squares and parks. Though rarely interfering with nightly pubbing/clubbing, organized crime is not absent, there have been brutal gang wars between Eastern European (Albanian/Russian) gangsters. Most of the riff-raff now is vegetating in Sankt Georg, which unfortunately begins on the central train station's western side. Drug-related crimes. Avoid the Steindamm area at night, be cautious even during daylight.
Hamburg also has some very ugly Ghetto outskirts very much reminiscent of the French banlieus. The worst one probably is Mümmelmannsberg in Hamburg's very east. The (quite nicely situated) districts of Rothenburgsort, Wilhelmsburg and parts of Harburg aren't much better.
When pubbing/clubbing in the Fishmarkt and Reeperbahn districts try not to separate yourself from the main crowds. There are dodgy corners -- avoid dark and empty spots.
80% of Hamburg's territory are very nice and very safe, including the entire Western and Northern parts of town.
St. Georg is neighborhood in downtown Hamburg undergoing a major regentrification process. The cleaning up of this area, however, is a bit slow in the coming for a good portion of the neighborhood, and the area immediately surrounding the central station still remains pretty dodgy (much more than St. Pauli's fabled red light district, The Reeperbahn, actually).
St. Georg actually begins at Kirchenalle, a street which is the second entrance/exit for Central Station, and a street which actually houses the main taxi stand, and a strip of tourist hotels, restaurants, and money exchange shops. If you stay on Kirchenallee, or to the museums and theatres surrounding you, you'll be fine. However, if this is your first time to Hamburg, or you're traveling solo without the benefit of a city native to show you around, try not to explore too deeply into St.Georg beyond Kirchenallee. At the central station itself, you will find the usual panhandlers and bums, but on Ballindamm (a cross street of Kirchenalle), you will find some pretty worn out looking prostitutes, drug dealers, and other dubious types.
Again, if you're traveling in a pack, or are willing to take a taxi into some of St. Georg's rennovated, newly hip areas for coffee and shopping, by all means go for it. But stick to the reccomendations you either receive from native friends or maybe from this website. Do not try to navigate the area by foot beyond a two block circumfrence from Kirchenalle or you might get lost, and probably turned off by what you see.
If you want to explore downtown Hamburg by foot, use the Möckenberg/Spitalerstarasse entrance/exit on the other side of Haupbanhof. It only takes you 3 minutes to reach it through the station, and the view is much "nicer" there.
Be careful in the area "Lange Reihe" St. George. There are lots of small restaurants, pubs and kiosks. if you are going to sit somewhere, make sure that you have taken all your your important stuff out off your jacket!! It already happend to me twice (stupid me), that i hung my jacket over the chair i was sitting on and some %//&!!" came, took the seat behind me an robbed my stuff while i was eating and talking to friends.
-I think there are some dangers in Hamburg like in other big cities.
Hamburg has arround 2 million people including the surburb and so there are some areas where you should be careful and aware.
I grew up here so I am not afraid in 'my' city but it is not advisable eg. to walk through the Schanzenpark or another park at night . (Although in the Stadtpark there are several people at night, having barbecue etc. and this is quite save)
Also some parts of town are very poor like St. Georg, Sternschanze, St. Pauli and do you should simply be careful like in every other big city.
St. Pauli is the redlight district and there are many tourists at night so it is quite safe too.
This picture shows Blankenese at the river Elbe. A nice place where the streets are stairs between the houses who are build on the hills.
Use some kind of secure carrying device for your wallet, passport, cellphone, and keys. These are critical items and it will really put a damper on your trip if any of these are lost or stolen.
I recommend some kind of safe carrying system, such as the Personal Secure Pocket: http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&ssPageName=STRK:MESELX:IT&item=140327196663