The common belief that posh is short for Port Out Starboard Home is a good example of a folk etymology. It sounds good, but it's just not true. The real story actually may be more interesting.
Posh is a loan word from Romany, the language of European "Gypsies". The Romany for Half Crown, a relatively high denomination coin, is posh karunna. British who came into frequent contact with Gypsies, mostly members of the underworld, adopted posh into their vernacular without really knowing that it only meant "half".
Other Romany loan words include shiv, which a direct cognate with the Romany word for knife, nark for police informer, from the Romany for nose (the term predates drug prohibition, so the common etymology of "narcotics officer" is unlikely) and "a rum go", a British expression for an odd occurence or thing. A rom go in Romany is a Gypsy (or, properly, Romany) thing.