You want to match at least one, so use '+' instead of '*' as the quantifier. Having passed that, you want to match a non-digit:

sub is_word_with_nondigit {
#    $_ = shift;
#    /^\w+$/ and /\D/;
    $_[0] =~ /^\w+$/ and $_[0] =~ /\D/;
}
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A different quantifier may be used if you want a minimum length, like /^\w{4,}$/

Update: Modified code to untaint the sub's argument, instead of checking a copy. If you don't like side effects use the commented-out version.

After Compline,
Zaxo