When BayAreaAC said "old water supply" - he meant "cold" water supply - typo - the water feed to the tank. Turning that off cuts of flow to the tank so will stop any tank leak - also to all hot water faucets, and also the hot tempering water that keeps the toilet tanks from "sweating" - toilets will still work, just may accumulate condensation on the outside.
About six things that could put water in the catch pan, and you should be able to track down which yours is with a flashlight and paper towel (to search for wet spots by wiping). Remember you are working with hot water and hot pipes, and also there is very hot exhaust gas going up the center flue - so don't burn yourself or set gloves or paper towel on fire.
1) dripping pressure relief valve - brass valve mounted near top of tank, with a pipe coming down to about 6 inches off the floor or deck the heater sits on. As these get old, they tend to drip a bit as the washer gets old. Rarely go bad quickly, but on occasion the drip can turn into a flow overnight, so not safe to leave tank pressurized while dripping. Not a safety hazard (other than that discharge is very hot) - just a flooding risk. Also, as they get older, the spring gets weaker, so it can sometimes operate at a lower pressure than it should - a false alarm, basically. If a leaking seal generally just a slow drip or few a minute - if has been operating tends to dump out about 1/4-1 cup of water per operation. Check if this is source by using paper towel on bottom of downtube to see if wet. Occasionally tis can be solved by openign the lever, which manually operates teh valve, for a second to flush it out and reset the spring seating, but usually once leaking needs to be replaced.
2) pressure relief valve operating with good cause, due to overpressure (or over-temperature, if pressure/temp valve used). This could mean a defective thermostat on the tank, temperature dial turned up too high (should be about 115-130), but generally means tank is boiling at the bottom, generating steam. Can be due to low water level, mineral buildup on the bottom of the due to corrosion that the gas flame overheats the water locally. If the relief valve has been operating it tends to dump out about 1/4-1 cup of water per operation, as it shuts off again (unless spring is corroded away) after overpressure is relieved. Check if this is source by using paper towel on bottom of downtube to see if wet.
3) Leaking valves or piping above or at the entry into the water heater - check by wiping pipes and fittings around top of tank with paper towel to check for leaks. Usually starts as a very slow dampness, working up to drip, then trickle, etc over weeks time. Commonly will be a lime buildup where the water is flowing and evaporting. Also check for leaks around electric elements (one or two, usually behind cover plate on front about 1/3 way from top and bottom) in electric heaters. NOTE - turn off power before removing covers. Also check around gas valve/thermostat housing on gas heaters, as the thermostat probe well into the tank can leak if corroded.
4) Leaking tank due to corrosion - can start as tiny drip and gradually work up, start as drip then rapidly progress to blowout that dumps the entire contents in a minute or less, or occasionally start right off as a total blowout. Would be dripping down inside the outer housing, usually through an insulation blanket, unless teh bottom is going out in which case drips down into firebox.
5) Leaking drain faucet at bottom of tank - check with paper towel at outlet (this is VERY common) as they use really cheap valves for this. Also check under edge of tank below where faucet goes in, as if threads are leaking at the tank it will drip out bottom edge there.If only due to dripping at the threads where you would hook a hose on to drain the tank, you can get threaded brass caps to put on there, with a hose washer (red rubber, high temp) and using teflon plumbers tape to seal the threads. DO not torque on the faucet hard - could break off where it enters the tank.
6) Condensate from combustion water vapor running down ducting into the flue through the center of a gas fired heater, dripping down into the center of the firing chamber. If has been doing for long, you will get a pile of rust there too. Usually will evaporate in the bottom of the firing chamber, but I suppose if that rusts through, could drip from there into pan. Can check by watching for drips from the duct through the venturi (open area right above top of heater) - commonly hear as a slash or sizzle as the drip goes into firebox. Note - a bit of loose rust in the firebox area is normal as gas heaters age, because the flue up through the center starts rusting, but big piles indicates a lot of drips coming from somewhere. NOTE - if this is a new event and you are experiencing the deep freeze effect, this could e due to condensation in the flue pipe which you would not normally see, if a very small amount.. You will get teaspoonfuls a day from this, not cupfulls.
Any leaking pipes, valves, fittings can be replaced - typically for about $100-200 total visit by a plumber. Leaking thermostat well can be fixed in some models - usually only leak in tanks about reaqdy to die anyway (usually 12-15 years, up to 20 in areas without corrosive or mineral rich water characteristics, which is pretty rare). Leaking tank itself is dead and has to be replaced - typically in $1000 plus or minus $250 range for normal size (30-50 gallon).