Well when I lived in Beirut in 2011, I first stayed in a hotel called 35 Rooms off Hamra and after we stayed in an apartment that we found through a friend. This year I lived in the AUB. It honestly depends on how long you are staying, where you want to stay in Beirut, and how much you want to pay. For some basics:
I suggest airbnb:
There is this facebook group called “Beirut apartments” that could also be helpful.
And honestly when you are walking around Beirut there are a ton of sign for flats for rent, and quite a few hostels that when you google they come up!