Did the Hanseatic League have any policy toward or involvement with Jews?
The cities of Hanseatic League belonged to different countries and a quick search indicates that they did not have a common policy with respect to Jews.
Of those countries, the most tolerant one to Jews was the kingdom of Poland, which later merged with the Grand Duchy of Lithuania, which evolved to Rzeczpospolita Obojga Narodów, which is translated as Polish-Lithuanian Commonwealth, but literally means Republic of the two peoples). These countries in general granted various rights to Jews (the policies varied with time), and had substantial Jewish polulation. In general the Rzeczpospolita was by far the most tolerant state in Europe with respect to religion.
On the other hand Jews were not permitted to live in most of the German and Russian controlled cities See, for example, http://www.jewishvirtuallibrary.org/jsource/judaica/ejud_0002_0013_0_12843.html and http://en.wikipedia.org/wiki/Jewish_community_of_Gda%C5%84sk
I cite from the second reference:
A Jewish community in Danzig proper did not exist until the 15th century. The earliest records indicate that Jews were present in Gdańsk as early as 11th century. In 1308 Gdańsk was taken over by the Teutonic Knights and a year later, the Grand Master of the Order, Siegfried von Feuchtwangen, forbid Jews to settle or remain in the city in an edict of non-toleration ("de non tolerandis Judaeis"). The knights weakened the restriction in early 15th century under pressure from the Grand Duke of Lithuania, Witold, and as a result a limited number of Jewish merchants from Lithuania and Volhynia were allowed to come to Danzig. Around 1440 a "Judengasse" ("Jewish Lane", modern Spichrzowa) and around 1454 a Jewish settlement existed.