Why 'Dutch courage'?

To have Dutch courage is to lose fear or inhibition by drinking alcohol.
It originally referred to a (purportedly medicinal) Dutch gin:
In 1650 Franciscus Sylvius, a Dutch doctor, created Dutch gin in an attempt create a diuretic medicine. This was then used by soldiers in the Thirty Years' War by English troops and was an instant success for its warming properties on the body in cold weather and its calming effects before battle. Source
The specific reference to gin later became generalised to all alcohol.

So it's refers to a specific Dutchman?

Yes, but anti Dutch sentiment can also be found in other phrases highlighting other alleged national shortcomings: meanness with money for example ('going Dutch' still means to pay for yourself rather than join in communal payment).

Why the anti-Dutch sentiment? 

On one level it was the result of great power rivalry - England (later Great Britain) fought no fewer than four wars between 1652 and 1784. There was also an ambivalent public sentiment towards England's Dutch king (William of Orange) who took the throne following the Glorious Revolution of 1698. 

Though on the surface 'King Billy' was hugely popular for 'saving' England from a Catholic accession, a foreign monarch will always inspire some mockery and resentment.

Dislike of the Dutch eventually faded. By the Nineteenth Century popular resentment was firmly redirected towards Britain's neighbours across the Channel.

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