I suspect people would be hesitant to let drunks in their car as drunks tend to throw-up. I'd twist this idea to something that is not uncommon in Australia. Often popular late opening bars have mini-buses that drive people home in the last hours or so until closing. Could you pool a few bars to fund (or local council funded) to do a booze bus that does this. Possibly have volunteer drivers but if a few bars chip in you could split costs to keep this low. Possibly have local council allow bars that contribute to this stay open an hour later type deal to incentive funding.

Another problem is people don't like leaving their cars to get the next day. In Sydney we have/had a service where small scooters come and drive you home in your own car (the scooter fits in the car boot): http://www.scooterangels.com.au/ It might be easier to get volunteers for this as you're not putting drunks in your own car + people get their car in the morning. Lie above you might be able to get some bars/council to fund if you want this to be free.

In general I'd expect self driving cars to fix this issue over the next 10 years or so.