Should I drink protein powder on days I don't workout?

It's kind of up to you, and how you go about your eating habits on non-workout days.

Everyone needs a certain amount of quality protein in their diet. Even if you aren't working out at all, protein is still important. In fact, your body recovers a lot of your rest days, so your nutrition is essential on those days as well. Also, remember that any physical activity can stress your body and use up extra protein. Although exercising increases your protein needs, even a lot of people who don't workout traditionally still find protein powders useful for meeting their requirements. 

Some people choose to only consume their protein powder on days that they workout, and if you would like to do that, then that's completely okay! I would just make sure that you are still meeting your protein needs on your days off/days with less activity by eating enough protein foods. Your protein intake should always come from a combination of different sources in your diet.