A coherent presentation of a monoid is an extension of a presentation of this monoid by a homotopy basis, making a natural cellular complex associated to the presentation contractible. In the case of Artin monoids, we show that the usual presentation defined by Artin, using braid relations, can be completed in a coherent presentation that we give in an explicit way. To be able to handle presentations that are not confluent, we develop a homotopical completion-reduction procedure that combines and extends methods of rewriting systems introduced by Squier and by Knuth and Bendix. Since any Artin monoid embeds in its Artin group, the coherent presentation of the monoid gives a coherent presentation of the group. In addition, the category of actions of a monoid on categories is equivalent to the category of 2-functors from a coherent presentation of the monoid to Cat. In this vein, our procedure gives also a new proof of a theorem of Deligne concerning the action of an Artin monoid on a category in terms of a presentation based on the Garside structure.