Learn French: En buvant son café, le patron voit le docker amarrer le navire. - While drinking his coffee, the boss sees the docker mooring the ship.

In French, when you use a verb of perception like voir (to see), entendre (to hear), or sentir (to feel/smell) to describe someone doing an action, that second verb must be in the infinitive. Even though English naturally says "sees someone doing", the French structure is "sees someone do", which gives us voit le docker amarrer.

What is the grammar rule behind En buvant?

En followed by a verb ending in -ant (the present participle) is a structure called the gerund. It is the most common way to express "while doing" or "by doing" something in French. Here, En buvant means "while drinking," showing that drinking the coffee and seeing the docker happen at the exact same time.

Does le patron mean a customer here?

No, le patron is a classic false friend. In French, it almost always means "the boss," "the manager," or "the owner" of a business. If you want to talk about a customer in a café, you should use the word le client.

If I wanted to replace "the docker" with a pronoun and say "The boss sees him mooring the ship," where would "him" go?

The direct object pronoun (le) goes right before the main conjugated verb, which in this case is the perception verb voit. You would write: Le patron le voit amarrer le navire.