I keep this bike outside and dont use it much. When I moved it today, rusty water
dripped out of the little pinholes in the seatstays and chainstays near where they meet.

What do I do to keep this bike from rusting from the inside out?

You could drill a small drainage hole in the bottom bracket where water collects.
There are anti-rust treatments for frames such as weiggle framesaver but I have had a lot of success with good old WD40.
Check that your seatpost and stem are not seized into place and smear some grease on the metal-metal contacts.

If that was my bike I'd pull the seat post and hang it upside down for a day or so to help the water drain and to dry out the interior.

Should I follow that up with WD40 treatment through all frame openings I can find?

I assume that is active, red rust that came out. I would strip the frame and kill the active rust with phosphoric acid. No use covering up active rust, it'll just keep eating away. The guys on the bike mechanics forum will have more suggestions.

knotty

Its chrome, so I don't really consider stripping an option.

I think Knotty meant to strip off all the parts, not the finish. FWIW, I would remove the parts, let it dry thoroughly, then coat and drain the interior with Rustoleum "Rust Reformer" or similar product that chemically changes the rust. You can follow up with a rust preventative after that.

Will that thicken the inside of the seat tube? I dont want to have to use a sledge hammer to get
the post back in.

Just a question. If this is a bike that you keep outside and don't use much, do you really want to do all that work in removing the parts and cleaning it up all good and proper?

Buy an aluminum frame,problem solved.

Its a beautiful chrome Bianchi Pista. I don't use it much, and I dont have room inside for it. But I love this bike.

No room inside? Put the big-screen TV outside. Seriously, in eastern Texas humidity, a steel bike left outside is a disposable bike.