The feeling of cold depends on many factors --- for instance, the wind, humidity, whether the person is standing in the sun or shade, whether he/she is exercising, the clothing worn, and whether his/her skin is damp.
Let's first look at how heat is taken away from the body. To simplify the argument, let's assume that the body is dry, exposed, motionless, in still air and in the shade. The question now becomes: whether dry air or damp air takes away heat from the body faster, i.e. whose thermal conductivity ('conductivity' from hereon) is higher.
Many people think that the conductivity is higher for moist air than for dry air. However, this is not true. Below is a curve obtained from experiments:-
The diagram above shows that the conductivity of water vapour is actually much less than that of dry air. So, if humidity (i.e. water vapour) has any effect on the conductivity of air, it would make it less conductive, not more.
So the simple answer is no.