After making an emergency run to fix a friend's broken pipes, I'm
thinking what can be done so this won't cost me much more time and money.

What broke were bathroom sink lines, just below the cutoff valves.
It's an old house, with little insulation. Full unfinished basements,
mostly concrete floors. Copper plumbing.

Pipe insulation. I'm not sure this has any impact. Does it?

Water dripping. Seems like a good idea, but these were dripping until
they turned to icicles.

So that leads me to thinking stopping basement drafts. The door to
this part of the basement was very badly sealed. How much tempering can
we get just by sealing basement air leaks? Is it worth the effort? The
house is huge. Should I even not worry about leaving some basement
ventilation in cold weather?

Jeff