My reading was that the Watcher's Council were paying a very considerable amount of money toward Buffy's upkeep as a slayer, just not by providing her with regular cheques. In the first 5 seasons Giles always has extensive access to a massive magical library, state of the art computers, training equipment, weaponry, etc etc all of which has to have been funded from somewhere. On top of that, they fund a suite for Faith and pay all of her expenses since she doesn't have a mother to live with.
In season 6, Giles puts a huge emphasis on Buffy taking on adult responsibility. Once the money from Joyce's life insurance runs down, Buffy is forced to get a job and actually pay some bills. During this time, she's at odds with the Watcher's Council and it's not like she wants to pick up the phone and ask them for cash.
At the end of Season 6 and the start of Season 7, she receives a cheque from Giles, then suddenly there are no further mentions of money. The Watcher's Council was destroyed, which presumably means that Giles just inherited all of their accounts and assets and given the urgency of their situation, he presumably just gives her money from that point onwards.