Using CGI.pm is quite fine. It is still a CPAN module, and still supported by Debian. Moving to PSGI/Plack might be an option, but it is not at all compulsory. In Debian Wheezy, CGI.pm V3.52 is part of system Perl (5.14), and there's also a package libcgi-pm-perl for Wheezy with version 3.61. In Debian Jessie, there's no CGI in Perl 5.24, but there's still the Debian package libcgi-pm-perl, now version 4.35. You might need to install that package after upgrading.

If you use CGI qw(:standard), then there's not much to fear, it is just the :any tag which is no longer available. A notable change is that if you call the param() method in list context, you get a warning: So many programmers have been bitten by failing to do this correctly, so now you get the warning even if you know what you're doing. If you deliberately chose to call param() in list context, you can suppress this warning by setting $CGI::LIST_CONTEXT_WARN=0.

You need to do something if you used the internal module structure of the CGI package, e.g. the Fh package, which no longer exists, and you might eventually need to do something if you are using the CGI::Pretty module.

So, if the task at hand is upgrading from Wheezy to Stretch then I suggest to just go ahead and leave your CGI applications as they are.