I know that making Sauerkraut is basically putting layers of cabbage and salt in a jar and then filling with water. However, I have been told that doing this at home can allow generation of good bacteria (probiotics) in the sauerkraut. How can I promote this growth while keeping the sauerkraut edible and tasty?
The ingredients of sauerkraut are very basic--its basically just cabbage and salt (the water is drawn out of the cabbage). Given this, you will produce the most nutritious kraut using high-quality cabbage and salt with natural minerals. A high quality sea salt will contain additional minerals that processed kosher and table salts lack (also, it is typically advised not to use salt with iodine--as it apparently prevents the growth of the desired bacteria). Additional ingredients such as spices will add their relative nutritional value to the mix.
Obviously this doesn't necessary maximize or preserve any "probiotic" aspect in particular, but as commenters have mentioned these claims are dubious as best.
Also, by the way, you only need to add water if the cabbage fails to produce enough brine to cover it after being salted. If you do add water you'll want to add additional salt so as to preserve the salinity--typically I've used 1Tbsp of salt per quart of water.
I usually leave fermenting vegetables out for about a week. Some variables in the final product are crunchiness and sourness, which I believe depend on time, temperature and the amount of salt used. You should taste your kraut a few times over the first few days to see how you like it, and when you think it's done then cover and put in the fridge to stop the fermentation.
By the way, you want your vegetables completely submerged to get the anaerobic effect, and you may see some foamy scum on the water surface which is harmless. You can skim and throw it away if you like, or just stir it down into the mix.