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.
They make both wall systems and flooring systems for wet basements. Sorry I am not posting links but don't feel like shopping today. A friend just put up wet basement panels (google that) and they look good and they are waterproof.
Whether this is the right solution for you depends on your house and how long you expect to be there. If your yard is graded poorly, your basement walls are in really bad shape, or there is some other intrinsic issue that will always cause moisture I would personally not do what Chris or bib suggested.
First it is a basement. How much are you willing to spend to make a place that naturally has water nice - there are "newer houses" and houses built in dry areas that don't have issues which are perfect to renovate - but that is not your house. I have seen many disappointed people spend thousands of dollars waterproofing basements to have their (newly finished) basement spring a leak the next year.
My suggestion is make the basement livable, safe, and enjoyable. Knowing that it could get flooded at some point. I would suggest putting in water-proof (rubber) planked flooring - HD sells this. Even if it is soaked it can be cleaned and reused. I would suggest some waterproof panels set about 4-6 inches from exterior walls. I would suggest no insulation other than rigid foam paneling. And if you do the foam paneling I would allow space for it to breathe. I would suggest neither drop ceiling or drywall for the ceiling - just paint it black or white (white you have to clean more).
If this extra space is really important to you - and you want it nice - I would suggest that you look at adding an addition to your house. After you do all of that waterproofing, foundation repairs, and finish your basement your $$$ is getting pretty close to adding on plus you don't have to worry about it being flooded.
Best Answer
There could be two issues: 1) moisture dripping off the building paper under the siding, or 2) moisture entering the masonry cells.
1) We install building paper (moisture barrier) under exterior siding because we know the siding will leak. Well, that moisture needs to run out at the bottom of the wall. If the siding is leaking really badly above this area for some reason, all that moisture could run out long after a storm and “wet” the wall.
2) If you look closely at the interior block wall, you’ll notice some discoloring (darker areas) aligned vertically. This indicates moisture is getting into the cells and filling with water. (Masonry blocks are hollow and often not filled with grout when the wall is constructed.) This allows moisture in the ground to seep into the concrete block cells. (Yes, masonry block is porous.)
Also, if the window in the first picture is the same window in the interior picture, then these moisture spots align perfectly. This could mean the cell is “full” of water.
Heat will “draw” moisture out of a wall. (That’s why paint on south facing walls blister more than other walls.) So, where the cells have moisture, the sun could draw the moisture out and cause the discoloration.
I’d: 1) check to see if moisture is running out from behind the siding by rolling a few paper towels up and stuffing them up under the bottom of your siding at the wet area AND at the dry areas. Then compare after a storm, 2) Dig down a few feet at the wet spot and see if the moisture barrier on the outside of the basement wall is damaged. I’d also make sure the material (soil) is porous so the moisture can “flow” away from your wall. If it’s not, I’d replace with drain rock. (Round river rock without fines...no rocks smaller than 1/2” dia.) Worse case: you’ll need to install a French drain to carry the moisture away from the wall.
Regardless of where the moisture is coming from, I would NOT finish the interior until you find and stop the moisture because it will cause mold.
...but I don’t understand what is happening at the top of your wall inside. Is that insulation?