No I wouldn't pay. I used to pay for Magpie, but that was in the 'old days' when there was no other way to recycle plastic. I expect all waste collections to be covered by my Council Tax.
And I really don't buy this 'no budget' line. The council have got 8 million quid to alter the road layout round Valley Gardens (whether that will be an improvement or not is a subject for another thread!) That apparently has come from some central pot from some government agency. Leaving aside the allocation of the Council's own resources are you really telling me that there is no central government or EU fund that couldn't have been approached to set up a garden waste recycling scheme? Brighton Council (and I accept it isn't 100% the fault of the Greens) simply had other priorities.
As far as I'm aware, grants tend to be for one off setup costs for projects. Even with the ridiculous Valley Gardens project the grant doesn't cover on going costs after implementation - something the council has flippantly suggested would come from local businesses stepping in and paying the costs.
So even if there is a fund for collecting garden waste it would only cover the setup costs which apparently are fairly low. Then there would be the ongoing £1m cost of running the service - where do you suggest that comes from ? How do CityClean find an EXTRA £1m when they are being told to cut £800k ? Why should people without gardens pay for the collection of garden waste ?