LED dimmer/controller: preserve state after loss of power?
You don't give any info about the LED strip, so I'll assume 12V and less than 2Amps.

The simplest solution would be to use a power supply which supports dimming by resistor, for example this one. Check out page 4. You need to order the correct version which is dimmable by connecting a potentiometer or a fixed resistor between the two DIM wires. It's really simple.

This kind of power supply behaves either as a constant current or constant voltage source. When using the dimming feature it will regulate current but the output voltage will not exceed 12V which is what you want. These cost about 20 bucks, which is pretty cheap and makes it not worthwile to roll your own.

If you forgot to mention that your LED strip is actually RGB, that won't work though.