It’s true, cloth napkins use a decent amount of energy and water for upkeep when you wash and dry them regularly. Plus the making of cloth napkins has an environmental impact, cotton is a water and pesticide intensive crop. However, if you’re buying new, you can choose hemp or flax, which don’t use as many resources. Secondhand is even better. I’ve gotten most of mine at secondhand stores and I’ve made some from used tablecloths. Unless you’re eating BBQ every night, chances are you won’t need to wash them after each use, cutting back on water and energy use. And cloth napkins last a long time, I’ve had some for over 20 years. Once they start to have too much wear, they make great rags.
Paper napkins can only be used once. They use less water and emissions to produce, but if you’re using one for each meal, that adds up. Paper napkins (and paper towels) can’t be recycled, but you can compost them.
And while cloth napkins have a bigger initial cost (especially if you’re buying new), since they last for years (and years), they come out the winner in the end.