Keep in mind: Every time you defy a resolution, you get 5

-"The world considers you a villain!" in every city that has the AP religion. That on top of war weariness can be really tough!
(It can be used against the enemy though

, if you have control of the AP and make a mental note on which resolution they are defying, you go for that resolution in every election and watch their 5

adding up.

)
NOTE: I don't play with 3.13 because of OOS-problems in multiplayer, I understand that 3.13 made some changes to the AP, so maybe the following advices are no good with 3.13. Hopefully people knowing 3.13 can jump in and comment/correct the advices.
My ways of dealing with the AP:
1. Build it myself, and spread my state religion to all my cities as they are founded/captured. That usually keeps me as president and thus in control of the resolutions.
2. If I don't build the AP, I keep an eye out for who did, and which religion the AP has.
2. a) If I have that religion, I spread it to all my cities and adapt it as my state religion, which makes me a full member with the possibility to be elected as president.
2. b) If I'm then not elected president, I'll vote "no" or "abstain" rather than defying (for defying it has to be something realy important, I've yet to find anything important enough to be worth the 5

).
2. c) If I do not have the religion of the AP, or I don't have enough votes (as a full member) to be elected president, I declare war on the owner of the AP asap. That owner becomes my no. 1 priority victim, heading straight for the AP city. (During that war, I may have to live with the occasional 10-turn-peace.

) Once AP is under my control, I spread the AP religion to all my cities, and adopt its religion as my SR. That normally does the trick of making me president, and the rest of the game (untill Mass Media) I can enjoy war or peace to my liking, never again having to worry about the silly votes.
3. And if I then spread the AP religion to all the other civs (one city each is enough), I might even get the diplo win.

Cheesey it may be, but since the AP is there, we have to live with it
EDIT: If I never get control of the AP, I gift/sell Mass Media to the owner asap, to make the AP obsolete.