rubber gasket under bathroom sink leak
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.