What \PassOptionsToClass{<option>}{<class>} does is pass its first argument as an option to the class specified by its second argument. Since \DeclareOption{<option>}{<code>} causes <code> to be executed whenever <option> is passed as an option, you can use \PassOptionsToClass here to pass on an option.
The following would work for you:
\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{beamer-nk}[2017/10/16 Beamer version nk, V4.1]
\DeclareOption{handout}{\PassOptionsToClass{\CurrentOption}{beamer}}
%% N.B., you could replace "\CurrentOption" by "handout" in the line above.
\ProcessOptions\relax
\LoadClass[xcolor=dvipsnames,professionalfont,french]{beamer}
If you also want to pass on other options to beamer you probably want the starred form, \DeclareOption*, which handles all unknown options (i.e., the ones you haven't declared).
\NeedsTeXFormat{LaTeX2e}
\ProvidesClass{beamer-nk}[2017/10/16 Beamer version nk, V4.1]
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{beamer}}
\ProcessOptions\relax
\LoadClass[xcolor=dvipsnames,professionalfont,french]{beamer}
\DeclareOption*{\ClassWarning{myclass}{Unknown option '\CurrentOption'}}safer ? – Tarass Mar 07 '18 at 17:35(Also: options to
– Circumscribe Mar 07 '18 at 18:54\documentclassaren't just passed to the class you are loading, but also to packages that can handle them regardless of whether the option is used by the class in question. A useful application of this is the draft option, which is used by many packages.)