The following code should fix the problem:
CreatePalette[Pane[InputField["Enter a string"]], WindowFloating -> False,
WindowClickSelect -> True];
But as we figured out it is not!
I read all available information about WindowClickSelect and WindowFloating options in Mathematica documentation.
I didn't find any notices that we can't use the options simultaneously.
I also didn't find any cautions that we can't use the options with Mathematica palettes.
Thus, such unexpected behavior is probably a bug in Mathematica.
I advice to contact Wolfram support team regarding this bug.