Circuit Analysis with constant current and voltage sources
I know that the answer is 24 volts by traversing from B to D to C to A
  adding up voltages of resistors, but should I not be considering the
  contribution of other elements in the circuit, for example, the 10 V
  source?


Yes it seems confusing but it isn't. If there's something uniquely in series with a current source then that "something" has no effect; the current source overrides its presence. So, you have 3 amps passing through the 8 ohm resistor and that produces a volt drop of 24 volts.

As previously stated, the 4 ohm in series with the 2 amp current source is irrelevant hence the voltage across the 10 ohm resistor is 20 volts and, this is cancelled by the opposing 20 volt source in the "B" line.

Hence, 24 volts is the final answer.