Your sink has cartridges for valves..they're under the handle. There should be a cap over the top of the handle (if you have crystal knobs) that can be removed to expose a screw that can be removed to expose the cartridge. If you have levers look for a small set screw on the side that can be loosened and the handle removed.
Personally, I'd inspect but of one was bad I'd suspect the water would come from one side or the other.. it's pretty rare they would both be bad (assuming you have two different handles). If you just have one then it's more possible.
You can take a channel lock pliers and gently tighten the retaining nut on top.
Cartridges can technically be replaced but may prove difficult to find given the quality we are dealing with here.
If all else fails, I'd just replace the faucet. They are the exact same as most residential faucets you can find at Home Depot, Lowe's, etc..
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007matman AKA Matt