Ultimately, how you clean your blackout curtains depends on their cleaning instructions.
Some blackout curtains can be machine washed in cold water, while others require handwashing or dry cleaning in order to maintain their effectiveness and keep the material from getting ruined. Be sure to check the instructions when you buy to make sure you know how to clean your curtains.
However, you can always use a lint roller or vacuum cleaner to clean off dust, pet hair, and dander from your curtain in between full cleanings. You can also generally spot clean your curtains with mild detergents, though be careful — some curtains may bleach or stain with certain chemicals.