holli is correct that a very straightforward way to search for text surrounded by parens, braces, brackets, etc., is to use the character class negation syntax [^...], as you want to stop capturing text to discard as soon as you find the matching parenthesis.

However, your original example has a couple of other pitfalls:

1. The '$' followed by "\)" is trying to match end-of-line followed by a parenthesis, which will never succeed.
2. The attempt to use the result of a match in a new regex (s/$1//) is not advisable in a couple of ways.  First, you didn't actually -capture- the parenthetical expression (remember, your escaped parentheses \( ...\) are matching the open/close parentheses in the pattern, so you would need an extra set to capture those: (\(...\)); otherwise $1 won't be defined).  Secondly, even if you are correctly capturing, it's not a good idea to immediately use $1 without making an assignment first, as any other regex match will change its value.  Thus, it's better to write something like:

       if($input =~ /^(\([^\)]*\))/)
       {
           my $pattern = $1;
           s/$pattern//;
       }
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3. But since you're already doing the work of searching for the pattern, why not just do the substitutition at the same time?:

       if($input =~ s/^(\([^\)]*\))//)
       {
           # Successfully trimmed (...) at beginning of line
       } else {
           # Didn't match ^(...)
       }
   [download]

Notice that the last code segment is essentially what holli has suggested for a regex: "if($input =~ s/^\(([^\)]+)\)\s*// )".  The 2 differences are that he trims any whitespace after the final ')' with "\s*", and his expression only trims what's in the parentheses if it's one or more characters, because of the '+'.  (If you want to match empty parentheses as well, use '*' instead of '+').