Hair dryers only work to produce heat if there’s power.
If there’s power, you will have heat and don’t need the hair dryer and the pipes will not likely freeze.
You can also buy pipe insulation and insulate your exposed plumbing. It works amazingly well at keeping the water in the pipes warm. They can lose a lot of heat traveling through a cold basement or crawl space .
They can be powered with small generators though. Most are around about 1800 watts or less. I’ve actually use one to thaw out outside pipes. We have a property in the country and a deep well that would sometimes freeze where the pressure switch mounted. Another thing that works well for frozen pipes, but you must be careful, is a small propane bottle with a torch attached to the top. They’re also great for starting a fire - works much better than matches!