Yes they were. Author and journalist, Barbara Smit,in her book 'Drei Streifen gegen Puma: Zwei verfeindete Brüder im Kampf um die Weltmarktführerschaft' (Three Stripes versus Puma) covers the cooperation of the brothers with the Nazi Party. Smit stated that both brothers signed business letters with a "Heil Hitler"; although Smit alleged that Rudolf was a bigger supporter of the Nazi Party than Adolf was.
After the war, the legend is that Rudolf Dassler believed that Adolf Dassler had told the American army that Rudolf was a member of the Waffen SS; Rudolf was 'picked up' by the American's and interrogated. Smit believes this is the more likely explanation for the rift between the brothers rather than the other legend: a misunderstanding in a bomb shelter when a member of Adolf Dassler's family said "the bastards are here again" when Rudolf Dassler's family entered the shelter; Rudolf believing the reference was directed towards his family rather than the RAF.
How committed the brothers were to the philosophy of the Nazi Party is unclear, it may simple have been a pragmatic business decision to join the party to ensure their company could operate with a minimum of bureaucratic interference. The Nazi boycott of Jewish businesses in Germany - began on the 1st of April, 1933 - proved largely unsuccessful, but it may have highlighted to the brothers the problems that could occur if your business was not viewed as being pro-Nazi Party.
The fact that their business - the Dasser Brothers shoe factory - came to national and international prominence due to their sponsorship of athletes of all ethnicity's - such as Jessie Owens - the brothers may have felt the need to prove their pro-Nazi credentials by joining the party. We can only speculate whether the brothers 'bought into' the racial and political philosophies of the Nazi Party.
After the war, the brothers founded their own factories (Adolf created Adidas and Rudolf created Puma) and a bitter rivalry continued until the death of both brothers in the 1970's.