Know your target audience and research what appeals to it. Figure out if your PSA is directed towards kids, teenagers, adults or all ages. Familiarize yourself with other PSAs targeting the same demographic or addressing the same topic.
Brainstorm the concept for your animation. Plot out what will happen and what kinds of characters will be involved and hash out a rough idea for a script.
Design characters in line with your concept. Think about making them the same age range and demographic as your target audience, since this will influence how your characters dress and act as well as the way the audience perceives them. Your message may be better received if the audience can relate to the characters. Do some test animations to decide how the characters will move and act in the final PSA.
Write a script, even if your PSA will not have any dialogue. Determine the sequence of events and how shots will be framed.
Draw up a storyboard based on the script, with rough sketches of the PSA's key events. Tweak the script and storyboard as necessary for a more cohesive final product.
Record the dialogue or music track. Final mixing can be tweaked later, but the audio component should be finished before the visual portion of the animation.
Using the storyboard as a guide, animate the PSA with your chosen method of animation. Hire an animation studio if needed, allowing the studio to bring your vision to life from the storyboards, designs and notes you provide.