Who knew ladies appliances would be so useful to mankind! :)
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 .