– Usually appearing in negative contexts, the phrase dry behind the ears means experienced, mature.
– Conversely, the phrase wet behind the ears means inexperienced, immature.

According to the Oxford English Dictionary (3^rd edition, 2015):
– The former phrase is a translation from German (noch nicht) trocken hinter den Ohren, attested in 1712 and meaning (not yet) dry behind the ears.
– The latter phrase is a translation from German (noch) nass hinter den Ohren, attested in 1642 and meaning (still) wet behind the ears. The variant (noch) feucht hinter den Ohren, (still) damp behind the ears, is attested in 1842.
– These German phrases apparently allude to the idea that the area behind the ears is the last part of a newborn’s body to become dry after birth.

The earliest instances that I have found of dry behind the ears and of wet behind the ears are from texts translated from German:

DRY BEHIND THE EARS

This phrase appears in Chapter 33 of Interesting Travels in America. Translated from the German of Bülow¹. For The Port Folio, published in The Port Folio Enlarged. By Oliver Oldschool, Esq. (Philadelphia, Pennsylvania) of Saturday 21^st August 1802:

Young Americans are for the most part, excessively silly company, the well educated and travelled persons excepted. The French call such inexperienced uneducated boys, green creoles, (des créoles verts,) as in German we usually say of such a person, “he is not yet dry behind the ears.”

¹ On Saturday 6^th March 1802, The Port Folio had explained that the author was

a German gentleman, by the name of Bülow, formerly an officer in the Prussian service, who in the month of July, 1795, embarked at this place (Hamburg) for the United States, where he had conceived the project of making a permanent settlement, and who, if we are rightly informed, is either already returned, or is expected every moment to arrive, being determined to live in his own country; convinced by his own experience, that Germany is quite as good as Pennsylvania.

WET BEHIND THE EARS

This phrase appears as wet behind his ears in Chapter 6 of The Two Students. From Horn’s Bohemian² Village Tales. Translated from the German for The Atlas, by J. C. Clegg, published in The Boston Semi-Weekly Atlas (Boston, Massachusetts) of Wednesday 26^th March 1851—Michael Kirchel, a clerk, wants to marry Milada, but she is in love with Jacob Pischta, a student; during a reception, Kirchel is confronting his rival:

This was too much. Jacob clenched his fists, and would have sprung at his adversary like a wild man, had not Johann, attracted by the noise, entered at the very instant, and held him back.
“Be still, Jacob?” said he, “let the Dutch³ Michael go—and you, Herr Kirchel,” continued he, turning to the latter, “you are here only as a guest, and if you wont [sic] be still you can leave! We students give the merenda, and as Jacob is a student, he has more right here than you! Come Jacob!”
He took his friend’s arm, and led him away, while Kirchel remained behind, pale with rage, and making such a noise that his companions shut the door to prevent it being heard in the street.
“Such a Bohemian zopak⁴ dares to tell me that? Such a louse student, who is still wet behind his ears, thinks because he is received in the castle, he is some great person! Let me alone, I will give him a box on the ear that will make him remember who is Michael.”

² Bohemia then belonged to the Habsburgs, the ruling family of Austria; it is now a region forming the western part of the Czech Republic.
³ Here, Dutch is used in the sense of German—cf. origin of ‘double Dutch’ and ‘High Dutch’ (‘gibberish’) and origin of ‘Dutch treat’ and ‘to go Dutch’.
⁴ The colloquial Austrian-German word Zopak was a pejorative name for the Czechs, derived from Czech copak, meaning what?.—[source: Specific language contact phenomena  in the Habsburg Empire and their possible utilization for teaching Czech as a foreign language in Austria, by Stefan Michael Newerkla, published in Studie z aplikované lingvistiky – Studies in Applied Linguistics (Charles University in Prague, Faculty of Arts, Czech Republic – 2014)]

IN FRENCH

The Revue des traditions populaires (Société des traditions populaires au Musée d’ethnographie du Trocadéro – Paris, France, March 1912), recorded the French phrase mouillé derrière les oreilles, wet behind the ears, used in Wallonia, the primarily French-speaking southern part of Belgium.

In July 1907, the Revue des traditions populaires had recorded the synonymous French phrase jaune derrière les oreilles, yellow behind the ears, used in Hainaut, a province of southern Belgium. It was explained as referring to the facts that young birds “have the base of the beak close to the ears” (“ont la base du bec près des oreilles”) and that their beak is yellow.