I am sure this is a very basic question, but I just want to be sure to not make any mistakes. I am using a transistor S8050 D331 to switch leds. So the base is connected with 1K resistor to the Arduino Due (3,3V) outputs. The emitter is connected to ground. The collector is connected to 1 led and a series 220Ohm resistor. The collector voltage is a seperate 5V supply.
Now the leds are too bright. What base resistor do I need to use if I increase the led resistor value? How are those related?
Increase this to 1k.
Do I have to alter the value of the transistor base resistor? I read about this and there seems to be a relationship that might be important.
No, don't. 1k on the base saturates the transistor nicely, don't mess with it. The relationships you mention, mostly dictate minimum values not nominal values.
No
Make the collector series LED resistor larger.
You can reduce the collector current by reducing the base current also, but that requires that you operate the transistor in its linear region. There, is not good for a driver because it requires the transistor to dissipate a lot more heat. It also forces the current to depend on transistor beta and thermally sensitive parameters, which makes the circuit unstable and unpredictable unless feedback is applied.
But is is a good idea to have the same value for the base resistor and the resistor on the led / collector side? 1KOhm in this case.
Why would it not be a good idea? Your reasoning, please...
One place to read up on using a transistor as a switch: https://www.electronics-tutorials.ws/transistor/tran_4.html
With a 1kOhm led resistor at 5V, I get approx 2mA current. If I assume the base current to be 1/10 of the collector-emitter current, I need an R = (3.3V-0.7V)/0.0002A = 13000Ohm resitor at the base side. So they cant be the same I think. But I might be wrong.
Why assume that? It gives you a maximum resistor value.
Because I read this a lot. Might be not correct. I am new to this.
"rules of thumb" are not sufficient to solve general problems. First try to understand how the devices work. Or just believe knowledgeable people...
The rule of thumb 1:10 gives you a maximum current ratio. Even when it's used as an estimate.
It reduces the number of parts in the BOM. You don't have to stock as many different values.
You want the transistor saturated in a switching application like this.
If you measure the Vce voltage and it is less than ≈ 0.7v (for a low power transistor) the transistor is saturated.
If a 1k series LED resistor gives a good intensity, that’s just fine.
Let’s say you increased the value of the base resistor; if the Vce ON voltage goes above .7v the base resistor needs to be decreased until the Vce voltage goes to .7v or less.
OK, I tried the 1kOhm resistor on a green 5mm led. It is still far too bright.
Then try 1.5k or 1.8k or 2.2k or 2.7k or 3.3k or 4.7k 
Please correct me if I am wrong: As far as I understand, for the switching application, it is important that the transistor is saturated, which can only be the case if the transistor base-emitter current is sufficient. So increasing the base resistor might block the current to much and the transistor will not fully switch. It also seems to be important that decreasing the base resistor must be done carefully, to not exceed the max pin current on the Arduino.
So basically, is there a region in which increasing the base resistor does not change the behavior of the switch as long a the base resitor is not too small (Arduinopin current safety) and not too large (not fully switched)?
How much time did you spend reading the link provided in reply #9?