Air has mass and anything with mass has weight.
It can be demonstrated by taking 2 identical ballons, a string, and a dowel.
Use the string and the dowel you can create a scale.
Now inflate one of the identical balloons and hang it on the side while the other ballon should be hanged on the other side of the scale.
You will notice that the ballon with air will be lower than the other and this explains that air has weight.