This is my personal view of what he is saying. I should say, I have never actually heard this phrase used by anyone before. Perhaps I don't mix in the right circles!
You mention that Jeremy Paxman has noted that the English possess "a strong natural gloominess".
To me, in this context his phrase "Every Englishman is born a double Scotch below par" means that every Englishman needs a double scotch in order to feel better about themselves and to bring themselves up to par, which in this sense, means up to the proper level. Par, is a golfing term which basically means "the standard or expected number of hits needed to sink your golf ball into the hole". This term has been applied to general life now, where things are either "above par" or better than expected, or "below par" worse than expected. So he is saying, Englishmen need a double scotch in order feel good about themselves and to be like other nationalities.
Strangely enough, it's quite a gloomy phrase itself!
Interested in Language