By basking do you mean the temp on top of a surface under a direct overhead heat source or do you mean the hot side temp on top of the substrate/on top of the glass (if you have a tank) or on top of the tub (if you have a tub)?

I don't believe it will be harmful to your BP to take her out to handle her if she is in her hot hide and your room is 73 degrees (my snake room is kept at an average of 76-78). That's not freezing by any means.

So no, the temp drop from her hot side to the cooler ambient room temps should not make your BP sick.