Depth of Wood Post for Clothesline
It depends, but 2-3 feet is likely ok

I have build many decks and fences and in doing so I have had to set a lot of posts. My standard response would be to a 4 foot hole with a 10 inch diameter construction tube but that would definitely be overkill. 

You need to figure out if you want to be able to remove them at some point, if that is the case definitely use construction (Sono) tube. Or if you plan on doing something else with the posts, ie flag pole, net for volleyball or badminton ... Also length of line will be a consideration.

Depending on the width of your footing I would say that anything in the 2-3 foot range is likely a safe bet but again remember there are many variables in play and it is always easier and safer to over build and you can be sure it will last forever. It never hurts to check local building codes, while you may not find much specially about posts for clothes lines you many come across something that helps.