Hello everyone!
Should I place pull down resistor near button? And if I should, what nominal it should be?
Yes, perhaps, and maybe. Depends on what you're using it for.
In some cases you don't need a resistor at all as what you're feeding may already have a pullup (or pulldown) resistor.
Case in point, the Atmel processors used in an Arduino, have a pullup you can enable with INPUT_PULLUP(aka pinMode(2,INPUT_PULLUP); to enable the pull up resistor. The only thing thing is you'll be hooking your switch from pin to ground instead, and your reading will show 0 when pressed and 1 when released, but that's easy in software.