You should be able to slow it down by caulking inside the sink around where the actual drain pipe comes in contact with the sink bowl. that's probably where the water's originating from.
No great answer for a temp fix. Clean existing seal and surrounding area with a non-abrasive cleaner such as Comet Classic spray-on. Be sure to remove any loose bits of rubber or other material around damaged seal before applying caulk.
Apply a generous bead of clear GE Silicone all around existing seal. Using a finger and gentle pressure, smooth the bead of caulk, pressing it into any gap or crevice around the seal.
Make certain you have complete coverage of any and all gaps. One you have applied this silicone, you cannot do a second application, as it will not bond. Remove excess with paper towel and allow 60 minutes minimum before checking efficacy. Be certain to used a new, unexpired tube of silicone caulk. Expired caulk will not cure.
Run water, check for leaking. After the caulk has cured, you may want to tape a short length of toweling all the way around the pipe. An old, clean rag will do. Check and replace toweling if it gets wet.
This won’t be pretty but it should keep the floor dry until you replace the sink and plumbing in January. Good luck.
Seals shrink over time with age. Therefore I would start by using a rench and tightening up the assembly. You might be done after that. If not the amount of water is that small that you just catch the water and do nothing else. Then I would disassemble the drain which is actually not much work and replace the gaskets. I don't believe that silicon is fixing this without disassembly anyway.
There are also replacement kits available for the complete assembly which don't cost much.