There might be a small piece of debris in stem under the handle.
The stem might be worn out and need replacement.
The stem under the handle may need lubrication after years of use.
Is the water still shutting off when the handle is used?
Has this cold water handle been getting worse over time, or was it a sudden occurrence?
Where are you located?
Have a basement under your home?
are you familiar with the procedure of turning off ALL the water to your house?
If you are, get your tools and meet me in the bathroom. lol