There is car parking at 203 Jay St, Brooklyn, NY 11201-1907 Edison park fast. Which is approx a 6-10 minute walk from the US District Court. (No street parking).
Visitors/family are not allowed at the oath ceremony itself but they can wait for you in the cafeteria upstairs.
Just so you know the travelling time for me was approx under 3 hours from Mastic Beach in Long Island and I left at 5:40am and walked in the doors after being searched etc at 8:30am, my appointment time was 8:28am. As a precautionary you might want to spend the night at a hotel or something nearby(closer).
• The following NYC subway lines stop in the vicinity of the Courthouse. The 2 and 3 trains stop at
Borough Hall and Clark Street. The numbers 4, 5 and the M and R stop at Court Street/Borough