First turn off the water leading to the bathroom sink. Open the spout and let the remaining water out. Next get under the sink and disconnect the elbow where the water goes down. Either get a new elbow or rechalk the old one with plumber glue or tape. Re-attach parts and turn water back and hope for the best.
Check for loose pipes or a bad gasket. Tighten any loose pipes you find or replace the gasket if it is bad. If the sink or pipes are cracked, then they may require replacement. A home repair book from Lowes or Home Depot will provide step-by-step instructions for making any of these repairs.
The best way at repairing a leak, when there is a leak, is to just replace the whole leaking pipe. It shouldn't be too difficult or expensive either. Go to the hardware store and buy the piece you need, and it's pretty simple to install.