Is there a passive component able to stabilize current (as zeners can do with voltage)?
What you are looking for is called a "constant current source". Unfortunately these are not passive components, however luckily they are available in 3 pin IC form that are very easy to implement into your circuit. 

Here is an example product: http://uk.rs-online.com/web/p/constant-current-diodes/5359098/

Datasheet: http://www.ti.com/lit/ds/symlink/lm134.pdf

From the data sheet implement it like this:

 


Connect the V+ pin to your supply voltage (it should be quite stable)
Connect the R pin to a resistor to V-, the current through the resistor proportionally controls the current out from the output. 
The output current is provided by V- and the current through the set resistor.


These parts are also available with fixed output currents hardwired into them. They can be slightly cheaper.