I noticed the water was dripping on my bathroom faucet... realizing it needs a washer... holding off getting someone to come in to fix it for me... I shut the hot water under the sink but continued to use the cold water. My washer needs to be replaced in the pipe of the sink, not the faucet head itself... something I don't know how to do... being female and plumbing challenged. I notice downstairs in my basement directly below this bathroom some water pooling up from the cement? Could this be because I shut one valve off and not the other and continued to use the sink... or could it be something else...

Could this be because I shut one valve off and not the other
No.

Post picture of faucet for assistance in replacing washer.