Yes, I would shut off the water, pry off the baseboard, and check it all out. Once you have it all out of there, you can clean that grout and tile floor really good before you put anything back together. If you can see that area from under the house, or from a basement, it would be a good idea to take a good look at it too, to make sure the floor isnt rotten under there.

Escutcheon?

It's called an escutcheon, and every hardware and big box store carries multiple styles and sizes. Get a plastic one that looks like chrome so it doesn't rust from condensation (or hopefully not a future leak).

If you are replacing the supply pipe, you can use a one-piece escutcheon. Just don't forget to slip it over the pipe before you install the rest of the components.

I second the suggestion to get a chrome plated plastic one, too. And replace the shutoff valve with a quarter turn ball valve.

It is also a good idea to operate the shut off valve every few months. Just turn it off and then turn it back on a couple/ three times.

Although I like 1/4 shutoff valves, there may not be anything wrong with that one. Just replace the escutcheon with the type in the photo. You don't need to replace the pipe, or do anything else other than clean things up. You don't need to make this a bigger deal than it is.

you can simply raise the current escutcheon up and snip it with tin shears or diagonal cutting pliers, clean the floor with vinegar and put your new escutcheon in place. Project done.

Cut that escutcheon out and replace with the pictured hinged version please. Leave the valve.

Thanks for all the comments. The valve might be 20 years old. is it okay to keep it?

Every time I have to work on plumbing with one of those valves like yours, I replace it with a quarter turn ball valve. I hate standing on my head and twisting and twisting the valve handle to operate it--worse under a vanity.

Angle stop valves are notorious for going bad over time. I live in a high rise building and the building has annual inspections where they test all of the angle stops in each unit.

If they aren't working properly then the owner has to fix them because in the event of any kind of leak or overflow not being able to shut off water to the toilet or sink can have catastrophic consequences for everyone below or even to the side.

My point is that if you are working on the plumbing fixture, use the opportunity to update everything since it is all accessible - water is turned off and the proactive fix will prevent future more expensive issues