Can I choose between Perl 6 multis that have no parameters?

You could destructure the | with ().

my $*DEBUG = 1;
show-env();

$*DEBUG = 0;
show-env();

# use an unnamed capture | but insist it has 0 arguments by destructuring
multi show-env ( | ()   where ? $*DEBUG ) { dd %*ENV }
multi show-env ( | ()   where ! $*DEBUG ) { True }

show-env(42); # Cannot resolve caller show-env(42); …

Or you could have a proto declaration

proto show-env (){*}
multi show-env ( |      where ? $*DEBUG ) { dd %*ENV }
multi show-env ( |      where ! $*DEBUG ) { True }

show-env(42); # Calling show-env(Int) will never work with proto signature () …

A more elegant way to insist that the capture is empty is to specify it with an empty sub-signature:

multi show-env ( | () where ? $*DEBUG ) { dd %*ENV }
multi show-env ( | () where ! $*DEBUG ) { True }