Best place for a switch when using an always connected power adapter
Power draw from a plugged in power adaptor that is not in use, is measurable, but negligible in the long run. A 2 Amp switching supply like that will cost you maybe a buck or two a year, if left unused but plugged in 24/7. Negligible is subjective. 

But the supply for that led strip, is just a power supply. It has no off mode, no flashing, nothing fancy. So unless you disconnect the strip, the supply will be powering the strip. Unless you disconnect the supply.

So you need to switch the outlet.

A simple method is a dollar store switched outlet adaptor.



Or a power strip.