Do you take a bath or have a bath?

Either is correct, but the idiom take a bath (or take a shower) is more frequently used in the US.
Take a bath and have a bath are both correct.

In older times, the term was "I need to draw a bath." which have stemmed to the more modern way of speech...

Neither one is wrong, but:

Most likely you will hear people say, "I need to take a bath." If you analyze what this is really saying it sounds like you are literally picking up the bathtub and taking it with you somewhere.

If you say, "I need to have a bath." It sounds a little more grammatically correct because you are going to go do the routine of having a bath by getting inside the bathtub.