Connected means associated with the mob.
A Connected Guy is vouched for by a “Madeguy, he is referred to as “a friend of mine”. Most important he can be “trusted” for his connections by other Wiseguys
Connected can also refer to relatives, friends of mobsters that are not criminals or businesses that pay a tribute to or are partially owned by a mobster. These people may not be criminals themselves but may benefit from their close association with a mobster, by receiving preferential or deferential treatment. E.g the brother of a mobster might be said to be connected even though he is not a member of a family or even a criminal. People would be wary of crossing him or disrespecting him because the respect accorded to his brother is also accorded to him in diluted form. In addition crossing a close relative or friend of a mobster can be seen as disrespect to the mobster, so people close to mobsters, even if not involved in the family or crime benefit from their connection with the mobster.
A connected business is a business that pays the mob a tribute ("protection money"). A connected business receives representation from the moster protecting them. This means they can't be shaken down by other mobsters, will be protected from other criminals and may receive other benefits such as access to corrupted officials in government or unions to reduce regulatory burdens. For example a connected business may be able to hire non union labor in a unionised industry, or receive licenses without conforming to licensing requirements.
In some industries it is or was nearly impossible to operate withut being connected, e.g. garbage industry in the New York city region.
In these cases the tribute may be paid to a trade organisation instead of directly to an individual mobster. E.g. the King's County Trade Waste Association which was a Genovese family run garbage "club".