Traditionally in household use, a cloth napkin was used a week. Hence monogrammed napkin rings, so you would get your own at hte table next time. This however is an aesthetic decision.
Another decision: How many times can you wash them before they start looking tacky?
A third decision: Starch: Starched cloth not only stays looking good longer, but it's easier to clean: Most of the stain soaks into the starch.
A final decision: Cotton or linen or polyester. Linen lasts longer but is pricier. Polyester can be perma-pressed so that it requires no ironing. Much higher stain resistance. Not as absorbent.
Labour: What is your labour worth to wash etc.
Let's look at year's worth of 3 meals a day for a family of 4.
Paper: 4 * 365 * 3 = 4300 napkins/year.
Vanity Fair napkins -- Amazon best seller -- 2 cents each. so $86.00 Nicer ones about double that.
Cotton cloth napkins $14/dozen. You can get them in colours, which helps to not show stains.
Washing: This is iffy. If you can include them in a load you are doing anyway, the cost of washing is essentially nil. Otherwise if you consider that a sheet (6x6 feet) is about 16 napkins (1.5 feet x 1.5 feet) but that napkins are several times as heavy a fabric. Lets call 1 sheet = 6 napkins. So a load of sheets is 8 sheets = 50 napkins. If you have a cost figured per load of wash, then you use 7 washloads of napkins per year (on a weekly cycle)
Starch and iron. Cost for this is vanishingly small except for your time. A friend runs a B&B and uses cloth napkins. I think he said it takes about an hour to do 50 napkins. Call it 1 minute each. If your time is worth $15/hour then that's 25 cents per napkin. I bet that is in line with the time it takes to wash dishes. Note however how quickly the time cost outstrips the original cost.
A 'slumming it' option is to not bother ironing them. Just fold them. If you get coloured ones, you can run them with the coloured tee-shirts. You're not having to make an impression. Likely in family use you would have an everyday set that you don't starch and iron, and a 'fancy guest' set that you make an extra effort for.
An even more slumming it option: Buy bulk rags. These are often sold my machinist shops for wiping oil up. Sold by the pound.
Conclusion: Worth doing, especially if you have a family.