Leaking under sink and st base on sink

Without seeing the leak and checking further I can't say yes or no.

You don't mention the brand but most faucet Mfgrs have rebuild kits.

follow this steps and fix it. God bless you
f a faucet leaks, replace seals and gaskets . If a shower head leaks, create a better seal or replace seals.
If a sillcock wears out, replace the washers .
If a compression faucet drips from the spout, replace the washer and/or the seat .
If a faucet leaks from the handle, replace O-ring(s), replace the packing washer, tighten the packing nut, or replace worn packing, depending on the valve type .
If a ball or cartridge faucet leaks from the collar, replace worn O-rings .
If a disk faucet leaks around the base or has reduced flow, replace cracked or pitted disk assembly or worn inlet seals .
If there is water under the sink, tighten the faucet-set locknuts under the sink. You also can replace putty or a gasket, replace a worn faucet, or replace leaky supply tubes
If the aerator leaks around the edge, replace the washer in the aerator
If the spray hose leaks or has reduced flow, replace the O-ring on the diverter valve, replace a worn washer at the base of the spray head, or clean the diverter valve and spray head

make this steps and fix it. God bless you
f a faucet leaks, replace seals and gaskets . If a shower head leaks, create a better seal or replace seals .
If a sillcock wears out, replace the washers .
If a compression faucet drips from the spout, replace the washer and/or the seat .
If a faucet leaks from the handle, replace O-ring(s), replace the packing washer, tighten the packing nut, or replace worn packing, depending on the valve type.
If a ball or cartridge faucet leaks from the collar, replace worn O-rings .
If a disk faucet leaks around the base or has reduced flow, replace cracked or pitted disk assembly or worn inlet seals
If there is water under the sink, tighten the faucet-set locknuts under the sink. You also can replace putty or a gasket, replace a worn faucet, or replace leaky supply tubes .
If the aerator leaks around the edge, replace the washer in the aerator
If the spray hose leaks or has reduced flow, replace the O-ring on the diverter valve, replace a worn washer at the base of the spray head, or clean the diverter valve and spray head

make this steps and fix it. God bless you
f a faucet leaks, replace seals and gaskets . If a shower head leaks, create a better seal or replace seals
If a sillcock wears out, replace the washers
If a compression faucet drips from the spout, replace the washer and/or the seat
If a faucet leaks from the handle, replace O-ring(s), replace the packing washer, tighten the packing nut, or replace worn packing, depending on the valve type
If a ball or cartridge faucet leaks from the collar, replace worn O-rings
If a disk faucet leaks around the base or has reduced flow, replace cracked or pitted disk assembly or worn inlet seals
If there is water under the sink, tighten the faucet-set locknuts under the sink. You also can replace putty or a gasket, replace a worn faucet, or replace leaky supply tubes
If the aerator leaks around the edge, replace the washer in the aerator
If the spray hose leaks or has reduced flow, replace the O-ring on the diverter valve, replace a worn washer at the base of the spray head, or clean the diverter valve and spray head

follow this steps and fix it. God bless you
If a faucet leaks, replace seals and gaskets .
If a shower head leaks, create a better seal or replace seals .
If a sillcock wears out, replace the washers .
If a compression faucet drips from the spout, replace the washer and/or the seat .
If a faucet leaks from the handle, replace O-ring(s), replace the packing washer, tighten the packing nut, or replace worn packing, depending on the valve type .
If a ball or cartridge faucet leaks from the collar, replace worn O-rings .
If a disk faucet leaks around the base or has reduced flow, replace cracked or pitted disk assembly or worn inlet seals .
If there is water under the sink, tighten the faucet-set locknuts under the sink. You also can replace putty or a gasket, replace a worn faucet, or replace leaky supply tubes .
If the aerator leaks around the edge, replace the washer in the aerator .
If the spray hose leaks or has reduced flow, replace the O-ring on the diverter valve, replace a worn washer at the base of the spray head, or clean the diverter valve and spray head .

Check the hose connections at the spray head nozzle it sounds like it is leaking there and running down the hose and ending up under the sink.

The first thing I would do is to locate source of the leak. Look carefully at faucet when it is on and see if you have any water leaking from areator and running back down the underside of spout and leaking around base of faucet, also see if you have any leaks around handle. If the leak is from the areator you can just replace it and not have to remove the faucet. when you check under sink on faucet supply lines make sure they aren't leaking at that point. I would use paper towels or rags to wipe everything dry under faucet and let it set for 1/2 hour then check again. If it.s still dry run some water and use both hot and cold water seperatly,checking underside in between. If you determine that leak is coming from top side and leaking thru. Shut off water supply valves if available if not shut off supply to house, make sure water is off before proceeding. Disconnect supply lines from faucet,Remove retaing nuts on either side of base,and remove washers. Remove faucet and check for any obvious faults. If you think it was leaking around base get a small tub of plumbers putty. Most hardware stores stock it. Remove a golf ball sized portion or larger and knead in your hands until it is warm and pliable. Press into place in faucet base. Continue doing this until it's full. Reinstall plastic or rubber base gasket and put faucet back in place Put washers and retaing nuts back on a tighten till snug. Center faucet where you want it and tighten securely. Scrape off excess putty with knife point and wipe with paper towels till clean. Hook up supply lines and turn water back on. Check for leaks and good luck. Thank you

well it do or it don't come out if it is old U R better off buying a new, it is the seals that R leaking, then it could B a weld gone bad