Old versions of Minecraft are unable to use the Mojang authentication system to confirm if users are premium users. This requires servers to run in offline mode allowing for cracked clients to connect to these servers. The issue can be addressed by bringing the previous authentication servers online or redirecting them.

Server Authentication:
Old URL: http://www.minecraft.net/game/checkserver.jsp?user=
New URL: http://session.minecraft.net/game/checkserver.jsp?user=

Client Authentication
Old URL: http://www.minecraft.net/game/joinserver.jsp?user=
New URL: http://session.minecraft.net/game/joinserver.jsp?user=

Due to these old URLs not working, illegitimate account holders can connect to any Beta/Alpha server.

The client authentication is the main URL that needs to be fixed, server owners can implement their own fixes to patch authentication server-side although it would be best to fix Server URLs as well.