Yes, they are connected. Just from Wikipedia about Season 1 (the 2012 series),
The first season aired in Japan between October 5, 2012 and April 5, 2013, with episodes 1-9 covering the Phantom Blood arc, and episodes 10-26 covering the Battle Tendency arc. The first part follows a young man named Jonathan Joestar who becomes involved in a battle against Dio Brando, who uses the power of a mysterious Stone Mask to become a vampire. The second part takes place fifty years later as Jonathan's grandson, Joseph Joestar, faces up against the Stone Mask's creators, the Pillar Men.
while for Season 2,
The second season, titled JoJo's Bizarre Adventure: Stardust Crusaders, adapts the series third story arc, Stardust Crusaders. Set in 1987, the story follows Joseph's grandson, Jotaro Kujo, and his companions as they seek to defeat DIO, who has placed a curse on his mother, using special abilities known as Stands.
as per the premise,
JoJo's Bizarre Adventure tells the story of the Joestar family, a family whose various members discover they are destined to take down supernatural foes using powers that they possess. The story is split up into unique parts, each following one member of the Joestar family, who inevitably has a name that can be abbreviated to the titular "JoJo".
you can see that the series are connected directly by the fact that they follow the Joestar family, and Season 1 and 2 are connected beyond that just the bloodline as on the Stardust Crusaders Wikipedia page
Joseph explains that the sudden appearance of Stands among the Joestar bloodline is caused by the nemesis of his grandfather, Jonathan Joestar: the vampire Dio Brando. Having succeeded in taking Jonathan's body after his assumed demise in the 19th century and resurfacing a few years prior, Dio has awoken his own Stand and recruited Stand-using assassins to kill Jonathan's remaining descendants.
