Adding the mechanical things, which are less glamorous than the biology and easier to fix.
Keep it submerged and keep the headspace small. Once the jar is half empty there is a lot of air over the top, and the surface behaves differently from the body of the jar. Moving the remainder into a smaller container when you are through half of it does more than it sounds like it should, and it is also when most people notice things going strange on top.
Press it down every time you take some. A clean utensil, and push the rest back under the brine. Two seconds.
Do not keep dipping the same fork in. Every trip introduces something, and by week three the population in the jar is not the one you started with.
On the practical answer to your actual question: make the same amount but in two smaller jars. Open one, leave the other sealed at the back of the bottom shelf. The unopened jar ages far more slowly than the one being opened daily, so you effectively get two week-one jars a month instead of one jar with a good half and a bad third. That single change fixed this for me and it costs nothing.