Slogans including "Irresponsible Politicians - Shame" were spray-painted in orange over the theatre's all-wood exterior, south of Calais.
The theatre's British architect Andrew Todd said it was an act of "politically-motivated desecration" by the far right.
Critics argue that the theatre is a waste of money.
Other slogans daubed on its exterior as well on walls in the neighbouring Chateau d'Hardelot included "Wart" and "The Debt".
"This was not the act of a bunch of teenage malcontents; it was a well-organised, adult, carefully-executed stunt relaying through violence the messages expressed already in the political sphere by the extreme right," said Mr Todd.
Situated about 30 miles (48 km) south of Calais, the Elizabethan-era theatre was commissioned by the Socialist-run Pas-de-Calais department at a cost of around €6m ($6.8m; £4.6m).  It is due to be inaugurated with a series of performances next weekend.
But it has drawn criticism from opponents of the local government, who say it is an elitist waste of money. The far-right National Front has some of its highest scores in the region around Calais.
With seating for 400 in balconies and a pit, the Hardelot theatre is the first in France to offer the intimacy of a Globe-style design. Several well-known French directors have welcomed it as a major innovation.
Mr Todd linked the vandalism to the "febrile atmosphere preceding the British EU referendum, in which Europe's far-right parties are attempting to stoke trouble".
But he said the building was "fundamentally an incarnation of Franco-British understanding, and would remain necessary whatever the result on Thursday".
France's first ever Shakespearean Globe-style theatre has been vandalised a week before its official opening.