Empty area below sink. Spread a sheet of newspaper flat over bottom of cabinet below sink. Secure with tape. If water drips from a leaking area, it will leave a wet spot on the paper. Place stoppers in sink(s) and fill with water. Leave sinks full of water for a while. Look for leaks. Remove stopper(s) and let sinks drain. Look for leaks. Turn on water and disposer and allow them to run while looking for leaks.
If leaks are not found, keep paper in place for a few days. If you see no indication of leaks, water may be from other sources. Check water filters, dish sprayer, faucet, dishwasher drain hose, sink/countertop seam, countertop joints.