Powering Arduino Pro Mini + Driving LED Strip
According to the schematics, the Pro Mini uses the MIC5205, which is a linear regulator.
Using it directly with 12V is not a very good idea, in terms of power efficiency, as it will dissipate 35mW vs 25mW provided to the ATMega board.

From this perspective, I'd recommend a step-down converter for powering the Arduino.

However it will not cause any damage, at least at room temperature, as the power dissipation is well within operating margins, see calculation at page 8 of the MIC datasheet.

Edit: detailed pros and cons, power wise.