Learn French: L'apprenti observe le souffleur de peur de rater le geste. - The apprentice observes the glassblower for fear of missing the gesture.

We use the infinitive because the subject is the same for both actions. It is the apprentice who is observing, and the apprentice who fears missing the gesture. If the subjects were different, you would use de peur que followed by the subjunctive.

Could I use de crainte de in this sentence instead?

Yes, de crainte de rater le geste means exactly the same thing. De crainte de is a direct synonym, though it can sound slightly more formal or literary than de peur de.

The translation says "glassblower", but isn't a souffleur just someone who blows?

Yes, it comes from the verb souffler (to blow). In a workshop setting, it is the standard shorthand for a souffleur de verre (glassblower). Fun fact: in a theater, a souffleur is a prompter who whispers forgotten lines to actors!

Does rater le geste mean failing to see the movement, or failing to perform it later?

It could be either, which makes rater a great word here! Rater is very common for both "to miss" an event or a target (like failing to see it) and "to fail" or botch a task, recipe, or physical action.