If I'm online, I'm usually there for trading or to register people as being online for Safari purposes. I'm not a battler, I don't raise competitive Pokemon, so for me, online battling's not really fun - I decline most people. I've thought about turning off requests for Passerbys and Acquaintances, but I've had fun trading with strangers and stuff so I'm hesitant to do that.
Most annoying to me was that my profile said for a long time "No battles plz" and I still got tons of battle requests! I ended up changing it because it apparently did nothing.
Editing to add that if you're looking for Friend Safaris, you should check around this forum or on Reddit. It's a good way to find the types you're looking for!