Well can't you just take out a fixed volume, say 10mL, and weight that?

If you want fancy continuous monitoring you need two pressure ports in the fluid, one at a certain height above the other one, and connect the two ports to a differential pressure sensor. Since beer is incompressible, you can then use the hydrostatic equation:

pressure difference = density x gravitational acceleration x height difference

to solve for density.

Look up how manometers work, it's pretty simple, but getting arduino to read the pressure will need some work.

-Z-