I mean, that entirely depends on the server design. Is it running one connection per process? One connection per thread? Presumably you've designed the server in a way that one slow connection doesn't block other connections from being handled fast, which means that this one connection whose handler went into the debugger is just now a really, really slow connection taking minutes (or hours!) to handle, but the other connections are running just fine.

Now, if your entire server is taken down because one connection threw an exception, that's bad design. But pretty much no major language works that way. All of them allow you to set things up so that an exception handling connection A won't affect connection B. And if you've done that in Common Lisp, then connection A halting and waiting for the debugger won't affect connection B either.