Not really enough info because most of the answer depends on what that little pipe is attached to behind that wall. Since you seem to be OK with R&R of baseboard and cutting into the wall, do it now and see what feeds the pipe.

The most likely stuff could be galvie iron, plastic PVC pipe, flex plastic tubing, rigid copper tubing, or flex copper tubing. Without being weirdly creative, that's a 95% chance of a bit of work once you're inside the wall.

Thanks Cold Weather.....it's probably not worth the effort to tear into the wall since it's really not that noticeable; just one of those little things that I don't like. Thanks for taking the time to respond.

How tall is the baseboard? I have the same situation with 6" high baseboards. The pipe just goes through a hole in the baseboard and there is a chrome flange that goes around the pipe to finish things off. Looks fine to me.

My baseboard is about 4 inches, just standard baseboard, but the flange that goes around it is half above and half below and I have to cut the baseboard to accommodate the lower part of the flange. Since the flange isn't as thick as the baseboard, it looks kind of weird. Thanks for responding.

If you have a grinder, you could cut away some of the back of the lower edge of the flange so that it matches the profile of the baseboard. Then the back edge of the top part of the flange will rest against the wall and the bottom part will be up against the profile of the baseboard. You wouldn't have to cut a large shape in the baseboard and it would look cleaner.

Thank you kudzu9. Unfortunately, I don't have a grinder, but I might be able to find someone who does and I have plenty of baseboard to practice on. That would certainly look better than it does now. I appreciate the suggestion.

You could install taller baseboard. Notch it, and fill the notch afterward. Or shorter baseboard (rip it down on a table saw).
Or omit the escutcheon and caulk the gap, then paint.

Thanks, homebound. Omitting the flange was a thought that came to me last night. I think that's really the only thing that's causing the problem. The pipe is located very close to the corner of the wall and I'm pretty sure it's just above the baseboard. Even though it doesn't show very much, it still bugs me. I'll take a closer look this weekend and, hopefully, removing it and caulking will work. Appreciate you taking the time to respond.