Go around putting coasters under their glasses. Hopefully they'll get the message.

I would just say pleasantly, "Would you like a coaster for your drink?" And then hand them one as you are saying it. NancyLouise

If they don't use them, then there really isn't much you can do. I feel like it would make them feel awkward if you asked them to use one.

Awkward guests or ruined furniture??
I vote for making an ill mannered guest know it.
Linda c

But I know you'd be nice about it, Lindac!

I think that Nancylouise's wording is perfect. If you need to deal with a whole group, you can say something like, "Here, let me get you some coasters" or "Oops, I forgot to put out the coasters -- here they are." You can put the coasters down under the drinks as you say it if you need to.

If you think you're going to have a group that won't get it, or it's a big party where people will be moving around a lot (I found plastic cups on the bookshelves in my bedroom after my son's rehearsal dinner!) you can put down tablecloths on wooden surfaces. You don't need coasters to protect glass, tile, plastic, Formica, or stone, just wipe up later.

I find the offenders are not those who are setting their glass down for a moment, but those who set the glass down and get distracted and walk away. That's when I walk around and slide a napkin or a coaster underneath the glass.
I have a friend who made a bazillion "glass panties"....or whatever you might call them. A circle of absorbent fabric with a top layer and sort of a slit to fit the stem of the glass into. Perhaps decorated with a ruffle or...not. All drinks are served in a footed glass....and no worries.

And yeah, Gelchom I hear you...I've found glasses under the edge of a sofa and among the nick-nacks on a shelf.
And after a late New Years Eve "breakfast" I found glasses that didn't even belong to me! Apparently people came from another party with their glass in hand.