The best and easiest solution:
spinner.getBackground().setColorFilter(getResources().getColor(R.color.red), PorterDuff.Mode.SRC_ATOP);
Other solution (Thanks to Simon) if you don’t want change all Spinners:
Drawable spinnerDrawable = spinner.getBackground().getConstantState().newDrawable();
spinnerDrawable.setColorFilter(getResources().getColor(R.color.red), PorterDuff.Mode.SRC_ATOP);
if (android.os.Build.VERSION.SDK_INT >= Build.VERSION_CODES.JELLY_BEAN) {
spinner.setBackground(spinnerDrawable);
}else{
spinner.setBackgroundDrawable(spinnerDrawable);
}