When you receive attention or a favor, acknowledge it by “I thank you,” instead of “Thanks.” We notice that “Thanks” has become a vulgarism from the abuse of the word.
—Mrs. Walter R. Houghton et al., Rules of Etiquette & Home Culture: Or, What to Do & How to Do it, 1893
Some people today still feel that thanks veers overly close to the informal, although the recommended alternative appears to be “thank you” rather than “I thank you.” Our definition for one sense of this word is “often used in an utterance containing no verb and serving as a courteous and somewhat informal expression of gratitude.”
The word has been used in this sense for over four hundred years; see, for example, a line uttered by Helena in Shakespeare’s All’s Well That Ends Well: “Thanks, sir; all the rest is mute.”