Hello - You can start by drying all the pipes with a towel, as well as the surounding floor or cabinet bottom. This will make it easier to locate the leak.
Does it leak ALL the time - does it only leak when you turn on the Hot water, OR the cold water.
Does it leak when water drains. ?????

Here's how I handle 90% of the "leaky sink" calls I get
1) dry everything (including my hands)
2) check for leaks on all pipes (drain AND supply)
3) Close the sink drain (or put in stopper, etc)
4) run the cold water - and watch for leaks
5) run the hot water - and watch for leaks
6) inspect the bottom of the sink for leaks
7) drain the sink and watch for leaks

If you find a leak in a supply line, examine it and replace or tighten as needed (not too tight) sometimes an inline "shut off" will leak, sometimes the faucet itself may leak from beneath.
If the leak is in the sink (cracked, etc) you'll need to replace the sink
If you find a leak at the base of the sink (where the drain hooks on) you may be able to tighten it or take it apart and replace the plumbers putty (you'll see it when you take it apart)
Leaks in the drain are most common. Metal pipes rot from the inside out and look fine until you grab them with a pliers and they crumble. Some times a joint in a pipe will loosen, or a pipe may be cut uneven and only 1/2 the pipe is inside the slip joint. In most cases I replace metal drains with plastic. It lasts and is easy to deal with.
Old bathroom drains can be a royal pain - so be patient and take your time.
If you need further help don't hesitate to let me know. Thanks. If this has helped please "accept"