Are there any districts I should avoid when renting an apartment for security (well, any other) reason? Common sense assumed.
Traditionally, Praga is considered the most dangerous districts. There are many dramatical stories, like that one that an estate agent have told me, after going there to see an appartment, he had to pay ransom to leave the staircase. But it don't apply to the whole Praga, only to a few streets.
As for general statistics, see here: http://dobraulica.pl/2013/03/przestepstwa-w-warszawie/
The highest chart are the burglaries per 10k citizen, as you see the worst are Włochy, and Ochota and Wola are bad too. Wola, together with Praga, are the places with the most murders.
But the districts are very diverse. Generally, the most safe are the new buildings, because there are only those living, who can afford it. Old buildings (from PRL times) are settled by children and grandchildren of people who have bought them, and many of them are unemployed, uneducated and without perspectives. You should inspect the vicinity carefully, watching for the groups of 'dressmen' (young people in sport clothes) sitting nearby. They can be really dangerous, and the areas where they settle are hazardous.
A very strong warning signal are also the gratings on the staircases, on the external site of the door, closed on extra lock. In more extreme cases, a fragment of the corridor with a few appartments has extra grating. Those little fortesses are a signal of very high criminality there (usually placed there, where you expect to be attacked when you leave your appartment, so the extra grating gives you the chance to see if the staircase is safe). If you see them, just find something else.
Look also for the signs of vandalism on the staircase. In safe buildings they are in good condition, in bad vicinities they look like slums.