cope with high variance or keep training
I put more training data. Now I use 70000 records instead of 45000.
My results:

precision: 0.81765974, recall: 0.65085715 on test-data
precision: 0.83833283, recall: 0.708 on training-data

I am pretty confident that this result is as good as possible. Thanks for reading
I built a neural network of the dimensions Layers = [203,100,100,100,2]. So I have 203 features and get two classes as a Result. I think, in my case, it would not be necessary to have two classes. My result is the prediction of a customer quitting his contract. So I guess one class would be sufficient (And 1 being quit, 0 being stay). I built the network with two classes to keep it flexible if I want to add more output-classes in the future.
I put dropout,batch_normalization, and weight-decay. I am training with an Adam-optimizer. At the end of the day, I come up with

precision: 0.7826087, recall: 0.6624 on test-data.
precision: 0.8418698, recall: 0.72445 on training-data

This means if I predict a customer to quit, I can be 78% confident that he really quits.
On the opposite, if he quits his contract, I predicted with 66% that he will do so.
So my classifier doesn´t work too bad. One thing keeps nagging at me: How do I know if there is any chance to do better still? In other words: Is there a possibility to calculate the Bayes-error my setup determines? Or to say it clearer: If the difference of my training-error and test-error is high like this, can I conclude for sure, that I am having a variance problem? Or is it possible that I must cope with the fact the test-accuracy cannot be improved?
What else can I try to train better?