Terms were not disclosed by the team, but the Salt Lake Tribune reported that the two-year pact was worth $18 million.
"Trevor is a seasoned big with a defensive mindset who will bring toughness, rebounding and a competitive edge to our team," Brooklyn general manager Sean Marks said in a statement.
Booker, the No. 23 overall pick by Minnesota in the 2010 draft, has averaged 6.5 points and 5.2 rebounds in 393 games over six seasons.
The 28-year-old played his first four seasons with Washington before signing with Utah in 2014. Last year with the Jazz, Booker averaged 5.9 points and 5.7 boards in 79 games.