How can I stop my basement from filling with water?
In many ways this is an easier problem to solve on a sloped area than a flat one.

First - waterproofing products applied to the inside of the basement are almost never an effective solution. The pressure of exterior water eventually pops them off the inside wall. On the outside they can work.

So, the solution is in two general forms - one is directing surface water away from the house (this involves making sure that the ground on the upslope side (actually, on all sides, including the upslope side) of the house slopes AWAY from the house for at least 10 feet/3 meters) so that water running on the surface of the ground is carried away from and around the house, not into it.

The other is to provide a drain to carry water away from the basement and out to a lower portion of the slope below basement level. The BEST (but most involved, generally) place to put such a drain is around the outside of the basement, below basement floor level. That involves digging up all around the house to place the drain (and anyone building a new house should remember that "when the foundation is being placed, this will only cost you the price of the pipe, since the hole is dug anyway.") Obviously a large project, also obviously the most effective solution. Water cannot get into the basement if it's carried away by a pipe before it can build up against the basement walls. You can also add a waterproofing membrane to the outside of the walls if you go this route, since the walls will be exposed, and you can additionally add a "drainage sheet" outside the waterproofing membrane to help get water down to the drain pipe. In most cases simply having the drain pipe and backfilling with stone or gravel should be adequate, since the stone or gravel provides the drainage path to the pipe, and the pipe means theres no way for water to build up against the side of the foundation.

If you cannot manage to dig a drain all the way around the house, you should at least run one from the low side, below basement floor level, out to a lower point on the slope where it can drain by gravity. You can knock a hole in the basement floor and install a sump for this to connect to, or not, depending on the water flow pattern in the basement. While there are plenty of people (such as the ones on a flat site who have little choice in the matter) who simply put in a sump and a sump pump, and pump the water out, if you have a slope available I'd strongly suggest the use of a naturally-sloped drain pipe to remove the water by gravity, without power or anything mechanical involved.