As Joev said earlier, charges could be applied for trash pickup and that could cover the cost of recycling. Properly funding allocation could also provide for the enforcement/cleanup of the small percentage of illegal dumping that could occur.
Alternatively, tipping fees at the city landfill could be increased, which would increase costs for any person or company dumping trash, incentivizing recycling as well as allocating additional funds for recycling programs.
This would be a good way to go about things, but free trash pickup is enshrined in the City Charter. A charter amendment is no easy feat.
A usage charge (like UA's per-bag sticker program) would be one way to go about things, but would also amount to a regressive tax on lower-income residents.
Perhaps next time we bail out a professional sports team or find ourselves with casino taxes the city should earmark them to something like a comprehensive recycling program.