We have a 100 year old house, and I would like to finish the basement but leave the stone walls exposed. I believe the stone can look great, but the mortar needs some work, and I am mostly focused on keeping the airflow to prevent any moisture problems. I'm aware that it is important to manage moisture outside the foundation. But when it comes to the inside, is it ok to somehow 'seal' the stone, or would that create an unwanted moisture barrier? In general, what is the best way to renovate the stone on the inside to make it look good without causing any other issues?
You can do it. Certainly.
There was a recent episode on this for "this old house". They basically suggested cleaning out the loose fill and then replacing the old mortar with fresh mortar. It looked tedious more than anything else - but doable by a homeowner. I've re-mortared and chinked stone foundations. It's a surprisingly high maintenance experience compared to poured walls.
The good news about stone foundations is that with proper bracing, you can fix up almost anything well enough to sell it to someone else who enjoys huffing dust, cobwebs, squirrel mummies and Radon Gas. But for the most part, it's like fight club - we just don't talk about it.
With improper bracing, you can experience the joy of being crushed to death, or lose an eye due to the magic of newtonian physics while you learn about shear loads and amateur metallurgy using those hydraulic jacks you bought at Harbour Freight. It's very educational.
A well built stone wall can always be repaired with less trouble than concrete foundations. The mortar and chinking is mostly for aesthetics, anyway.
Sealing the walls is fine, as long as you do it on the outside first, and then and only then do the inside. You'll have to dig down to the foundation to do this, and that bracing is an issue on the outside as well.
Now the part you won't like. I did this in high school for a relative who had a nearly 200 year old house with a sandstone & rock foundation, in a damp (flood plain) area. It looked great, but ultimately it was wasted effort. Exposed rock and mortar foundations will always breath, sweat and disappoint anyone expecting a modern basement experience. You can apply gallons of sealer, but it will simply create areas where it perforates.
These foundations were designed to keep out animals, hold up your house so it doesn't rot all that quickly and very little else.
They will always be home to spiders. You can seal it with gunnite or epoxy and it will still allow infiltration. It might look OK, in that "I love Halloween" fashion so popular with the gothic/serial killer/lonely artist set, but it will never be a comfortable space.
You might build a second wall that can be plumbed, framed and sheet rocked, but it won't solve much over the long run, even with substantial vapor barriers and ducting. The walls will still slowly shift, moisture will come and go, and you'll never be happy with the low ceilings.
So in summary: You can redo the walls. You can seal them. You can even box them in and add ducting, insulation, AC and lighting. It's still going to be a basement that weeps moisture due to temp and pressure differentials.
I fixed up the stone foundation of my 1890's house by picking and then with new mortar years ago. Now I am getting to using drylock (which has proven to hold the mortar together better than when left bare) and building walls for hookups.
There isn't much to do for insulation, unless you have the money to buy two-part spray foam kits ($600+). Whatever you do to seal the dust and make the place habitable is good. Just remember to use pressure treated wood against concrete and do everything like it will stay there for 50 years.