Our house was built in 1905 and the concrete floor is uneven and gets a lot of moisture at times. We replaced the gutters which helped a little. We'd like to finish the basement, maybe use stained concrete. Is newer concrete normally better, less prone to getting as moist? Is there an additional way to create a moisture barrier that wasn't used in 1905? I would consider the moisture to be relatively minor, but too much to consider installing a floor or carpet.
Concrete is porous, regardless of whether it's new or old. If you replaced it with a new floor, you could have a drain installed around the perimeter to stop most of the moisture infiltration. The best solution is to install a barrier around the perimeter of the exterior, which requires a lot of digging to get to the bottom of the foundation and installing a drain at the bottom to remove the moisture that gets to this barrier. A less intrusive solution is to install a French drain anywhere you have moisture problems around the outside of your home.
Regardless of how you decide to fix this problem, you should ensure that the yard is properly graded away from your property on all sides so that any moisture in the ground goes somewhere other than against the concrete foundation. This, combined with your work on the gutters, could be enough to make the moisture levels tolerable.
As for finishing, after you address the drainage issues, consider using DriCore or a comparable product as your subfloor before any finish materials go in.
DriCore is a composite subfloor material consisting of a dimpled heavy duty plastic bottom and OSB top. Each panel is 2'x2' and they install in a tongue-in-groove fashion. The point of this subfloor is that if you get a small amount of moisture on the concrete even after waterproofing, the dimpled bottom allows water to drain and with proper venting installed at points around the perimeter, air flow can get underneath to allow the remaining moisture to evaporate.