One possible answer in IMHO would be to combine some of the alternatives already presented here:

Heater system distribution + working area scans

Make your gigantic city be located on a frozen planet or any other place that is cold for the whole year.

Government supplies hot air/steam to all houses to keep inhabitants alive against the cold. Drug is pumped in the ventilation system's central distributor.

Those who work for the government get the antidote everyday on the DNA SCAN at the door to the building where their workplace is.

I really liked the idea of distributing by air or some other central system, you would only need a small crew to insert the drug at the distributor.

Whoever suggested the DNA scans, +1 to you. Gattaca all the way.