To the wall, yes, but not the floor. Any gaps that may remain are covered with shoe molding.

Just to be clear, I am not concerned with appearance with this question. The issue is air-sealing.

Where is the air coming from? Is this over the garage or something?

With the baseboard off, fill the openings that are allowing the draft. You can tuck strips of insulation into the gaps, and/or backer-rod, etc. Then install the baseboard and shoe molding. Caulk would look like heck and dirt will stick to dried, unpainted painters caulk.

The separate garage has problems of its own. The house s a 70-year-old raised house with a lot of problems that won't be fixed with a little backer rod, cauk or expanding foam any time soon. Suffice to say that I cant get to the source from "here". The best I can do unitil I am ready to spray foam the underside of the house is leat it leak or, maybe, try to stop some of it from the inside.

What I am considering is caulking/sealing on the hidden parts of the baseboard, like the back side where it meets the plaster.

It should not cost much to buy a tube of caulk.

I would think it would be easier/better and a nicer finished "look" to stuff the gaps when you have the baseboards removed. You could even buy a roll of rim joist gasket and tear it to the sizes you need and stuff it in place with a paint stick. It's not hard or expensive. Good luck with your decision.

caulking the top of the molding to the
wall and the bottom to the floor..is what
I do when moldings are not removed.

but if the moldings are not installed
then caulk the plaster to the floor.
(plaster walls wood floors do I remember
this right? )
I find the seal is better, because you don't
have to worry about the look..the molding
will cover it up.

if the gap is over 1/4" wide..backer rod is
easy to cut a bit oversized and use flat
screwdriver or butter knife to push into
crack. then caulk both sides of backer rod
to wall/floor. I usually cover backer rod with
caulk too. (control freak..I admit it!)

if you caulk moldings to the floor, and have
pets..the hair & caulk make it difficult
to clean. learned that the hard way.

I use Dap's brand Alex crystal clear caulk
with 25+ year life. goes on white, dries clear.
then when caulk is dry..inspect it carefully
to see that there are no gaps caused by shrinkage
of caulk ad it dries.
I also buy contractor pack of caulk.
always a use for caulk in a stick framed
house on piers.

so you have plans to foam insulate the floors?
I've been thinking of doing the same.
any idea as to who you'd use?

best of luck & Merry Christmas Ionized.

Thanks for the responses, everyone.

ER, I have not picked a spray foam person yet. I will have to do a lot of plumbing and electrical before that is done. There is probably a bathroom renovation to do first as well.

I understand. I have to replace bathroom floor
re-locate sink & toilet & upgrade plumbing
to kitchen sink. thinking about using pex.

once that list is completed...I can get ready to
foam.

one project ALWAYS leads to another..and another...

have a Merry Christmas everyone!

While we are at it, we might as well-- . Expensive words.

I get it. My cousin would say not, "while we are at it", but, "don't touch that".