Suppose that we are modeling the log-hazard of death using age, gender, bmi, and weight. BMI and weight are correlated. Would it make sense to drop one of these variables from out analysis?
Only if the correlation is very high (at least above .9) should you drop before even fitting a model. You should also try models with just BMI, just weight, and then both variables. See how the coefficients change. I would first start with both variables in and then move forward with a variable selection technique if needed.