I have a router's power supply. 12 Volts is written on the label. When I use a multimeter the voltage is 16.80V. Should I use it to power a small LED strip that requires 12V?
I trust the meter, and I assume the power supply is a 'wall-wart' style encapsulated blob which provides a few tens of watts at most. (A picture would certainly help.)
Also, since you didn't specify conditions, I assume you measured the output of the power supply with no load.
My guess is that it's a loosely-regulated one which drifts high as the load decreases. Read the label and apply the rated load (i.e. if it's 12V / 2A - apply a 2A load) and see what the output voltage is.
The suitability of this power supply for your LED strip is going to depend on a number of things, including how much current the strip is going to need. If the supply is rated higher than what the strip needs, the voltage will likely be higher than 12V and you may get more current than you bargained for (brighter LEDs maybe).